MUREG/CR-1 96ORNL/NUREG/TM--371
RAGTIME: A FORTRAN 1VIlmpementatfon of a
TimeoDependent ModelR Radio nuclides
in Agricultural Systems
First Progress Report
J, C. PleasantL. M. McDowell-BoyerG, G. KilIough
Prepared for theU.S. Nuclear Regulatory Commission
Division of Safeguards, Fuel Cycle,and Environmental Research
Under interagency Agreement DOE 40-5n50-75
Printed in the United States of America. Available fromNational Technical Information Service
U.S. Department of Commerce5285 Port Royal Road, Springfield, Virginia 22161
Available from
GPO Sales ProgramDivision of Technical Information and Document Control
U.S. Nuclear Regulatory CommissionWashington, D.C. 20555
This report was prepared as an account of work sponsored by an agency of theUnited StatesGovernment. Neither the U nited States Government nor any agencythereof, nor any of their employees, makes any'warranty, express or implied, orassumes any legal liability or responsibility for the accuracy, completeness, orusefulness of any information, apparatus, product, or process disclosed, orrepresents that its use would not infringe privately owned rights. Reference hereinto any specific commercial product, process, or service by trade name, trademark,manufacturer, or otherwise, does not necessarily constitute or imply itsendorsement, recommendation, or favoring by the United States Government orany agency thereof. The views and opinions of authors expressed herein do notnecessarily state or reflect those of the United States Government or any agencythereof.
NUREG/CR-1196ORNL/NUREG/TM-371Dist. Category RH
Contract No. W-7405-eng-26
Health and Safety Research Division
RAGTIME: A FORTRAN IV IMPLEMENTATION OF A TIME-DEPENDENTMODEL FOR RADIONUCLIDES IN AGRICULTURAL SYSTEMS
FIRST PROGRESS REPORT
J. C. Pleasant*L. M. McDowell-BoyerG. G. Killough
*East Tennessee. State University
NOTICE
This document contains information of a preliminary nature.It is subject to revision or correction and therefore doesnot represent a final report.
Manuscript Completed: December 1979Date Published: May 1980
Prepared for the U.S. Nuclear Regulatory CommissionDivision of Safeguards, Fuel Cycle, and Environmental Research
Under Interagency Agreement DOE 40-550-75NRC FIN No. B0188
Prepared by theOAK RIDGE NATIONAL LABORATORY
Oak Ridge, Tennessee 37830operated by
UNION CARBIDE CORPORATIONfor the
DEPARTMENT OF ENERGY
CONTENTS
Page
LIST OF TABLES ............. ........................... v
HIGHLIGHTS ................. ............................ vii
1. INTRODUCTION ............................. ......... 1
2. DESCRIPTION OF THE MODEL ....... ..... .................. 3
2.1 Radioactivity Transfer to Crops ............ 3
2.2 Radioactivity Transfer to Beef and Milk ..... ........ 6
2.3 The System of Equations ............ ................ 7
3. PARAMETERS DESCRIBING TRANSFER TO CROPS .......... . . . . 15
3.1 Time-dependent Aerosol Interception (s$ and S2) ..... .. 16
3.2 Time-dependent Root Uptake (Tpt.)i ............ 18
3.3 Time-independent Parameters (T e,s' T t's, p,h) ..... 19
4. PARAMETERS DESCRIBING TRANSFER TO BEEF AND MILK .... ....... 21
4.1 Contamination of Pasture Grass ..... ............. ... 21
4.2 Contamination of Beef and Milk ..... ............. ... 23
5. SOLUTION OF THE SYSTEM OF DIFFERENTIAL EQUATIONS .... ...... 29
5.1 Use of the GEAR Subroutine ..... ............... ... 29
5.2 Use of the Bateman Equations as a Check ... ....... .. 33
6. THE RAGTIME CODE ........... ...................... ... 37
6.1 Input .............. ......................... ... 37
6.2 Logical Structure of the Code ..... ............. ... 39
6.3 Subroutine CHECK ........ .................... ... 47
ACKNOWLEDGEMENT .............. ......................... ... 47
REFERENCES .................. ............................ 49
APPENDIX A. LISTING OF THE RAGTIME CODE ..... ............. ... 51
APPENDIX B. SAMPLE RUN OF THE RAGTIME CODE ....... ........... 93
iii
LIST OF TABLES
Table Page
2.1 Description of symbols used in the RAGTIME model .......... 11
5.1 Definition of RAGTIME state variables in terms of GEARpackage notation ........... ..................... ... 30
5.2 Definition of RAGTIME derivatives in terms of GEARpackage notation .............. ..................... 31
6.1 Radionuclide-dependent input parameters for RAGTIME ... 39
6.2 Nuclide-independent input parameters for RAGTIME ..... ... 40
6.3 Conversion factors used to convert compartmental concen-trations to total activity ...... ................. ... 44
v
HIGHLIGHTS
RAGTIME is a FORTRAN IV program that calculates radionuclide con-
centrations in food crops, beef, and milk which are contaminated as a
result of deposition of radioactivity on an agricultural area. Con-
tamination of these foods is assumed to occur as a result of the depo-
sition of radioactivity onto the surface of above-ground food crops,
pasture grass, the soil surface below crops, and the soil surface or
root mat below pasture grass, with ingrowth of radioactive daughters
being computed explicitly. The input source of radioactivity may be
prescribed by the user as a step function for each nuclide in the
chain. The model employs time-dependent interception fractions for
deposition of activity on food crops; the interception fractions for
deposition on pasture grass or pasture soil are at present constants,
but the facility for use of time-dependent values is provided. Season-
al aspects of the transfer of radionuclides between various compart-
ments of the model include the provision for specifying the dates of
emergence and harvest for various crop categories.
The system of differential equations describing the model is
solved by use of a discrete-variable numerical integration (the GEAR
package), and the accuracy of this solution is monitored by comparing
the total radioactivity in the system as calculated by the numerical
procedure with that calculated by use of an explicit solution of the
Bateman equations.
This report discusses the development of the model which is pres-
ently on-going, and thus, does not represent the final version envi-
sioned for implementation. Output for a sample run of the current
version is provided in this report.
vii
1. INTRODUCTION
A number of terrestrial food-chain transport models have been
developed over the past several years for use in assessing ingestion
dose to man from aerially deposited radionuclides.' Included among
these is a transport model described in the U. S. Nuclear Regulatory
Commission's (USNRC) Regulatory Guide 1.109, a model which NRC consid-
ers acceptable for assessing terrestrial transport of radionuclides
released during normal operation of light-water-cooled nuclear power
plants. 2 Limitations of these models with respect to dynamic seasonal
considerations as well as radioactive daughter ingrowth prompted model
development work, the initial progress of which is reported here.
RAGTIME (Radionuclides in AGricultural systems: a TIME-dependent
model) computes radionuclide concentrations in food crops, beef, and
milk which are contaminated by radionuclide deposition. The model
assumes a known rate of deposition of radioactivity (microcuries per
square meter per day, pCi m-2 day-') at a given environmental location
and uses interception fractions S1, S2, S3, and S4 to calculate radio-
activity input rates to the model compartments representing above-
ground food, the soil surface below the food crop, pasture grass, and• . TML is
pasture soil or root mat, respectively, at that location. RAGTIME s
basically an adaptation of the previously developed TERMOD code3 to
consider both seasonality of agricultural processes and the dynamics of
daughter ingrowth of radionuclides during food-chain transport. Be-
cause the development of RAGTIME is still in progress, some parameters
and concepts believed to be inadequate for the intended use of this
code have been carried over from TERMOD until appropriate revisions can
be made.
The system of linear ordinary differential equations describing
the model accounts explicitly for ingrowth of radioactive daughters and
provides for an input source of each member of a radionuclide chain.
This system is solved by use of the GEAR package 4 for solution of sys-
tems of ordinary differential equations. A subroutine, CHECK, of RAG-
TIME provides a check on the accuracy of this solution. At each output
I
2
time, CHECK makes use of an explicit solution of the Bateman equations
to calculate the total amount of radioactivity in the system; this
value is compared with the total obtained by summing the amounts of
radioactivity in all model compartments as computed by the GEAR sub-
routi ne.
In Sect. 2 of this report, we describe in broad outline the RAG-
TIME methodology and present the equations which describe the model.Sections 3 and 4 provide details concerning 4Le interception fractions
S1 , Se,, St, and S4 and the transfer coefficients of the system. Sec-
tion 5 is devoted to a discussion of the use of the GEAR package in
solving the system of differential equations of the model and to a
description of the way the Bateman equations are used. Finally, a
description of the RAGTIME computer code is provided in Sect. 6, giving
details regarding input, logical structure, and calls to the GEAR sub-
routines as well as the procedure employed by subroutine CHECK to moni-
tor the accuracy of the numerical solution.
As mentioned previously, there are a number of limitations of the
present version of the RAGTIME model which remain to be addressed.
Principal among planned refinements is the inclusion of the seasonal
cycle in the dairy and beef pathways. Aside from an example run for
the 9 °Sr- 90 Y chain, this report will not present a data base, the
development of which is presently in progress. The present code uses
an array of output times with a fixed size; whereas the integration
interval may be of indefinite length, this fixed array limits the out-
put density which is possible without recompilation. It may be desir-
able to remove this dependence on a fixed array in a future version.
Among planned revisions is the conversion of the RAGTIME code to
the International System of Units (SI), thus effecting a change in the
expression for radioactivity from curie (Ci) to becquerel (Bq), where
1 Ci = 3.7 x 1016Bq. (1.1)
The current version of the code uses Ci to represent activity, and the
following documentation is consistent with this convention although the
output may easily be converted to Bq using the relationship given above.
2. DESCRIPTION OF THE MODEL
The RAGTIME model is represented schematically in Fig. 2.1. The
subscript i associated with the compartments Ei, Si, Pi, etc. refers to
the ith nuclide of a radionuclide decay chain. Certain of the transfer
coefficients are nuclide-, or element-, dependent; this is also signi-
fied by the use of the subscript i, [e.g., (Tprt)i]. The deposition4 1k , 4-,•uu,• ~ ~ ~ -,he Inp•=, •, ,,ut source of r•adioac I. . . .I,source rersn I Uu sorc L, auuLIViy corresponudinq LoI -
the ith nuclide of a radionuclide chain. This source strength may vary
with time and may be represented in the computer code as a step func-
tion for each nuclide in the chain. The fractions of input radioactiv-
ity which are intercepted by above-ground crops, soil surface below the
food crop, pasture grass, and pasture soil are represented by S1, S2 ,
63, and S4, respectively. These fractions may be time-dependent with
respect to growth dynamics of the crop land or pasture. In Sect. 3.1,
we describe models for calculation of time-dependent interception frac-
tions S1 and S2. The present version of the code uses values of S3 and
S4 which are constant with respect to time; however, the same subrou-
tine is used to return values for all interception fractions, so that a
convenient method is available should the user desire to prescribe
time-dependent values for S3 and S4.
A general outline of the terrestrial pathways considered in RAG-
TIME follows, along with a brief description of parameters used in
implementing the computer simulation. A more rigorous definition and
the rationale behind the particular quantifications used for each
parameter are given in Sects. 3 and 4 of this report.
2.1 Radioactivity Transfer to Crops
Radioactivity deposited on the surface of the above-ground food
crop passes to the soil surface below the food crop with an environmen-
tal half-time of usually less than 30 days. 5 We have used 14 days for
this value ET[e = In 2/(14 days) = 0.0495 day-']. 3 , 6 For transfere,s
3
4
ORNL-DWG 79-19516R
• PLANT• -T
-. o . .o.. .
* *
• .
CROP HOLDLCOMPARTMEI
(EH)i (,uCi/kg
g,b i
BEEF MILK IBi (,uCi/kg) C(CIiQ)
JP BEEF HOLDUP MILK HOLDUPNT COMPARTMENT COMPARTMENT
(BH)i (/gCi/kg) (CH)i (WCi/h) I
M (Ci),-I
(COMPARTMENT FOR MASS BALANCE CHECK) I
Fig. 2.1. Schematic representation of radioactivity transfer tofood crops, milk, and beef as simulated by the RAGTIME computer code.
5
from the soil surface below the food crop to the subsurface soil pool,
we have assumed a 1000-day environmental half-time, givinig T =s,pln 2/(1000 days) = 6.93 x 10-1 day-. Radioactivity in the subsurface
soil pool is available for uptake by plant roots. The plant interior
compartment Ti simulates radioactivity which is transferred to the
edible parts of crops as a result of root uptake.
In Sect. 3.2 and Appendix B, we describe a model for calcultion of
a nuclide-and time-dependent rate coefficient (T representing this
transfer of activity. We have assumed a loss rate of 4% per year from
the subsurface soil compartment Pi to the soil compartment below the
roots, Hi, giving Tp,h = 0.04/(365 days) = 1.096 x 10-4 day-' (Refs.
3,7). The dotted lines from compartments Ei (surface of above-ground
food crop) and Ti (plant interior) to (EH)i (crop holdup compartment)
represent harvest of crops.
The level of radioactivity in all compartments associated with
crops at a given time is dependent on the histories of both the deposi-
tion source strength and on the growth of these crops. The effect of
crop growth upon the activity level on crop surfaces (compartment Ei)
is simulated through use of the time-dependent interception fraction
S1 . The time-dependent transfer coefficient (p't )i serves this func-
tion with regard to the plant interior compartment Ti. Before the
emergence of plants, the value of S1 is zero, as is that for (Tp't)i.
At harvest time, the entire food crop is assumed to be stored in a
holdup compartment (EH)i, after which time the radioactivity concentra-
tion level in this food is assumed to be affected only by the radio-
activity decay process. Thus, the activity level in the compartments
representing crops in the field (compartments Ei and TY) is zero except
at times between the emergence and harvest of crops. Harvest of crops
is simulated numerically by reinitialization of the state variables
representing these compartments at harvest time, [i.e., the compart-
ments Ei and Ti which represent activity associated with crops in the
field are set to zero, and the compartment (EH)i representing harvested
crops is assigned a value (in microcuries per kilogram) to reflect its
receipt of all radioactivity from Ei and Ti].
6
2.2 Radioactivity Transfer to Beef and Milk
As in the case of transfer from the surface of the above-ground
food crop to the soil surface, we have assumed a 14-day environmental,
or retention, half-time for the loss of radioactivity from pasture
grass to pasture soil [r = In 2/(14 days) = 0.0495 day- 1 ]. 3 ' 6 Tog,r
account for uptake of radioactivity by pasture grass from soil, we have
assumed a transfer rate of 1%o per year from R. to Gi [Erg = 0.01/(365
days) 2.74 x 10-s day-']. 3' 7 As in the determination of -p,h' a
transfer rate of 4% per year from the pasture soil compartment Ri to.
the soil below the roots Di [. ,d = 0.04/(365 days) = 1.096 x 10-4
day-'] was assumed.3, 7 The rate coefficient for loss of activity from
pasture grass resulting from grass consumption by. a cow is denoted by
Tg, . The derivation of a value for this coefficient is discussed in
Sect. 4.1. The beef compartment Bi represents the concentration of
activity in the muscle of a steer and Ci simulates the concentration of
activity in milk in the udder of a cow. It is not assumed that the
total loss from the pasture grass compartment, Gi, due to a cow's grass
consumption is accounted for by gains to the beef and milk compartments
B. and C. Rather, the transfer coefficients ()g,b)i and (Tg,c)i
account for only portions of the total activity transferred to the cow
through consumption of grass, those portions being the activity trans-
ferred to beef (Bi) and milk (Ci), respectively. The remainder of the
loss from Gi due to a cow's consumption is considered only for the pur-
pose of allowing a mass-balance check of total radioactivity in the
system. This remainder, being the complement of (Tg,b)i and ('g,cdi
with respect to Tg , is indicated in Fig. 2.1 as a dashed line drawn
to the compartment M.. The compartment M! is used only in connection1 1
with the performance of a mass-balance check. Details concerning the
procedure used by the code to perform this check are given in Sects.
5.2 and 6.3.
The dotted lines from the beef and milk compartments B. and C. to
the holdup compartments (BH)i and (CH)i, respectively, represent the
effect of storage on the radionuclide concentration in these foods. At
7
each output time, the computer prints, in addition to the activity con-
centrations of beef (Bi) and milk (Ci) at the given time, the concen-
tration levels which these foods would reach if stored for a user-
specified period of time (th for beef, th for milk). Thus, at a givenI b coutput time t, (BH)i represents the activity concentration (pCi kg-')
of nuclide i in beef at time t + tbh which was stored from time t to
time t + t , assuming a concentration level Bi at time t. The defini-
tion of (CH). (,c, i+ter-) is similar. Since the determination of the
values of (BH i and (Cn)i from those for B. and C. involves only the
application of the process of radioactive decay (using the Bateman
equations), the system of differential Eqs. (2.1) through (2.11) repre-
senting the model depicted in Fig. 2.1 does not contain equations cor-
responding to these holdup compartments. In Sect. 6.2 we present
details concerning calls to a subroutine, RESDNS, which uses an explic-
it solution of the Bateman equations to calculate values for (BH). and
(CH)i•
2.3 The System of Equations
The following system of equations describes the transfer of de-
posited radioactivity to food crops, beef, and milk as depicted in Fig.
2.1. As pointed out in Sect. 2.2, the compartments (BH)i and (CH)i are
not represented by differential equations since their values are cal-
culated using only the Bateman equations (see Sect. 6.2). Furthermore,
the differential equation for the compartment M', which is used only in
connection with a mass-balance check, is not included here but is dis-
cussed in Sect. 6.2. Definitions of the compartments used in the RAG-
TIME model follow the system of equations. Descriptions of all other
quantities used in these equations along with the values used for
certain of those which represent constants (at present) are given in
Table 2.1. Values of nuclide-dependent and crop-specific parameters
for a sample run of RAGTIME are given in Appendix B (Table B.1) along
with a description of how these values are derived from empirical data.
8
Crop surface
dE. - R+ )EdE1 SFi(t) _ (x. + Te )Eidt 1 ,s
i-i
Sj=1(2.1)
Crop soil surface
dS.dt 1 S=Fi(t) + E - Rh + TSIP)si + NR
i-1
j=l(2.2)
Crop soil pooi
dP.
dt rAPSi
Pasture grass
XR + .01 +Tp't )i Tp,h i+
-ij
j=1(2.3)
dG.~dt -=S 3 Fi(t)- R + g,r 91 ý)Gi T r,g i+
R-
j=1
B..G.B- .G
(2.4)
Pasture soil
DR.OR S4F (t) + T Gdt i •g,r i
- R + + )R + XR1 r,g r,d i 1
j=1
(2.5)
Pasture soil sink
dDti R di Ddt - •r,d R i X D
i-i
+ ij B.Oj=1
(2.6)
Milk
dC.1 -T (•~ )iGi - (XR + Ti)C +
g1Cl 1 milk
i-1xi • Bi jCj
j=1
(2.7)
9
Beef
dB.dBi= (Tg )iGj -
dtI -( g,b i i -
Interior of crops
dT.1 (Tp t)P. _
[X R + T + (•e )i]Bi + XR1 bee + exc 1 1
i-1
Bi jBjj=1
(2.8)
i-1
R1 Rx •T + X- S] B .Tj=1
(') ~*~\~ ZI)
Crop soil sink
dH. iR +( h/A)P - XHn + XR B. .H.dt -( p, i ]ih
j=l
Stored cropsd(EH) R Ri-1
1 - X (EH) + X Bij(EH)dt i 1 1 j 3
j=1
The RAGTIME compartments are described as follows:
E. Radioactivity present on the surface ofground food crop per square meter ofwhich the crop is grown (pCi m-2 )
(2.10)
(2.11)
the above-surface on
S. Radioactivity present at the soil surface belowI food crops (pCi m-2 )
P. Radioactivity present in the subsurface soil poolassociated with one man's food supply (pCi)
G. Radioactivity present in the pasture grass compart-1 ment (pCi m-2 )
R. Radioactivity present in the pasture soil from1 ground surface to the root depth of the grass
(pCi m-2 )
D. Radioactivity present in the pasture soil below the1 root depth (pCi m-2 )
10
Ci Concentration of radioactivity in the milk (pCiliter-1 )
B. Concentration of radioactivity in beef (pCi kg-1)
T. Radioactivity present in the interior of plantsproduced for human consumption (pCi)
H. Radioactivity present in the crop soil below theroot depth (pCi m-2 )
(EH). Concentration of radioactivity in food which isstored following harvest of crops (pCi kg-")
(BH)i Concentration of radioactivity in the beef holdupcompartment (pCi kg-1 )
(CH)i Concentration of radioactivity in the milk holdupcompartment (pCi liter-')
11
Table 2.1 Description of symbols used in the RAGTIME model
FORTRANSymbol Name Description Typea
A A Soil surface area (m2 ) required tofurnish food crops for one man
A ASUBG Pasture area per cow (m2 )g
Bij B(I,J) Radioactive branching ratio from bspecies j to species i (j < i)
B. BSUBIV(I) Concentration of nuclide i per b
1v unit fresh weight -in plant (pCi
kg-') divided by concentrationof nuclide per unit dry weightin soil (pCi kg- 1 )
d SMALLD Depth of plow layer (cm)
D DSUBG Dry weight areal grass densityg (kg m-2 )
Fi(t) F(I,T) Input rate (pCi m-2 day-') for b, cithnuclide at time t (days)
(Ff)i FSUBF(I) Fraction of the daily intake of bnuclide i by a beef cow whichappears per kg of flesh at timeof slaughter (day kg-1 )
(Fm)i FSUBM(I) Fraction of the daily intake of b,nuclide i by a dairy cow which
appears per liter of milk atequilibrium (day liter-')
xR LAMRR(I) Radioactive decay rate for ith b1 nuclide (day-')
Mb MSUBB Mass of muscle on a steer at timeof slaughter (kg)
p RHO Density of the soil (g cm- 3 )
S1 S1 Interception fraction for surface C, dof above-ground food crop
12
Table 2.1 (continued)
Symbol
S2
S3
$4
Tbeef
T e,s
T excdi
FORTRANName
S2
S3
S4
TAUBEF
TAUES
TAUEXC(I)
TAUGB(I)
TAUGC(I)
TAUGR
TAUMLK
TAUPH
TAUPT(I)
Description
Interception fraction for soilsurface below food crop (1 - Si)
Interception fraction for pasturegrass (0.25)
interception fraction for soilsurface or root mat below pasturegrass (0.75)
Fraction of the beef herd slaughteredper day (day-')
Transfer coefficient from Ei to Si(day-')
Excretion rate of stable isotopeof the nuclide from the muscle ofa steer (day-')
Rate coefficient representing lossof radioactivity from pasture grassdue to cow's consumption of grass(day-'); defined to be Vc/(A Dg)
Transfer coefficient from G. to B.(Mi2 kg -i day-i) 1
Transfer coefficient from G. to C.(m 2 liter-' day-') l
Transfer coefficient from Gi to R.(day-') 1
Transfer rate of milk from the udder(day-')
Transfer coefficient from Pi to H.(day-') 1
Transfer coefficient from P. to Ti(day-') 1
Type a
c, d
b
b
b
bc, .d
T , *
CC g,b)i
( g,cdi
g,r
Tmilk
Tp,h
(p,t)i
13
Table 2.1 (continued)
FORTRANSymbol Name Description Typea
T TAURD Transfer coefficient from R. to D.(day-') I
Tr g TAURG Transfer coefficient from Ri to Gi,(day- 1 )1 1
LSIP TAUSP Transfer coeficient, from S., to(day-')
th TIMBH Holdup time for beef (days)b
th TIMCH Holdup time for milk (days)c
U U Milk capacity of the udder (liters)
Vc VSUBC Dry weight consumption per day bya cow (kg day-')
aNo type specified means parameter is nuclide- and time-independent.
bNuclide-dependent.
C Time-dependent.
dCrop-specific parameters necessary to derive these quantities are
described and quantified in Appendix B of this report.
3. PARAMETERS DESCRIBING TRANSFER TO CROPS
Modes of contamination of edible portions of crop plants include
the interception and retention of aerially depositing radionuclides by
crops as well as root uptake following deposition onto crop soils. In
addition, interception and retention by crops of radionuclides resus-
pended from soil may contribute to the contamination of foodstuffs,
although this pathway is not represented in the model at present. The
relative importance of each of these modes of contamination will depend
on many factors including the mobility of the radionuclide in soil, the
availability of the nuclide for root uptake, the radiological half-life
(-lives) involved, as well as the types of crops being considered.
In developing a model to describe these pathways of contamination,
it is ncessary to consider seasonal cycles of crops (i.e., when the
crops are and are not present during the year) as well as the time
dependency of parameters describing contamination. Much of this time
dependency is due to physiological and morphological changes in plants
due to growth and maturation, and to changes which may be occurring in
the chemical form of the radionuclide which is deposited either on
plant or soil surfaces. Although time dependency due to chemical
transformations has not been considered to date, the present model can
potentially incorporate parameters of this type. At present, the time
dependency of the interception parameters S1 and S2 (see Fig. 2.1) and
of the root uptake parameter (Tp t)i have been studied, and preliminary
approaches to characterizing the dynamic nature of these parameters are
discussed below.A holdup compartment, (EH)i, is utilized in this model in order to
account for radioactive decay and daughter buildup that may occur
between harvest of crops and consumption by man (Fig. 2.1). The length
of this interval is left to the user's discretion.
15
16
3.1 Time-dependent Aerosol Interception (S1 and S2)
The interception of airborne radionuclides by edible portions of
crop plant will depend on these major factors:
1. the surface area exposed to depositing particles,
2. the shape of the edible portion and its orientation to
depositing particles, and
3. the particle density of the depositing material.
Because some of these parameters are specific for different plant
species, edible crops have been divided into categories representing
plant parts with similar morphological characteristics. The following
categories have been recognized:
1. root crops and other crops with protected edible
parts,
2. leafy vegetables,
3. exposed-grain crops,
4. cylindrical vegetables, and
5. spherical vegetables.
Ideally, empirical data regarding interception of airborne particles
over the growth cycles of crops should be used to represent the time
dependency of this mode of contamination. However, because data of
this type are not readily available, certain simplifying assumptions
were made to account for the factors affecting interception. It is
hoped that the adequacy of these assumptions will be tested at a later
date through validation studies and sensitivity analyses.
At present, we have attempted to model the dynamics of intercep-
tion by assuming that interception is a direct function of the project-
ed surface area of the edible portion, and that interception occurs at
a 900 angle to the plane of the projected surface area. The time
dependency of interception thus relies on the relationship of the pro-
jected surface area of the edible portion to the mass of that portion
during plant growth. This relationship may be characterized either
through the use of empirical data - for example, that obtained by the
Stanford Research Institute9 for several crops - or by assuming the
17
density of the vegetative matter remains constant during growth and
using geometric procedures to derive projected areas as a function of
plant growth.
The mathematical representation of the crop compartment Ei, which
intercepts depositing radionuclides, was given in Sect. 2.3 [Eq. (2.1)]
as
dE. i-ii_ .. . R + Te )E + R..Edt •Ii• ^ si i ijr "
The fraction s1 of depositing radionuclides intercepted by the edible
portion may be defined in two ways, depending on whether this fraction
represents an empirically derived or theoretical (geometric) relation-
ship with time. An empirical approach is
$1 = Som1L)w , (3.1)
where
o and nL = empirical constants,SL L
m = time-dependent mass of the plant part (g dry-weight),
w = number of plant parts per square meter.
An example of a time-dependent curve for m is given in Appendix B.
This representation of 51 is adapted from a document prepared by the
Stanford Research Institute, 9 which supplies values for the empirical
constants S, and n L for certain plant species. The empirical constants
were derived for considerations of specific surface area (mi2 g 1 )
alone, for a given planting density w, and thus may be appropriate only
for similar planting configurations. A geometric approach for estimat-
ing the relationship of S1 with time is provided by Miller, 9 where the
change in projected surface area with time may be calculated by assign-
ing a geometric configuration that best approximates the plant part of
interest.
18
The output for a sample run of RAGTIME, which considers grains in
compartment Ei (surface of above-ground crops) and is listed in Appen-
dix B, was obtained by using the empirical approach for estimating the
time dependency of -1. Empirical constants, used in Eq. (3.1), are
also available for edible parts of a few other plant species, including
beans, cabbage, peppers, and squash. 9
The value of S2, shown in Fig. 2.1, representing deposition of
airborne radionuclides onto the soil surface below the food crop (S.),is calculated h\b ascsming that
S2 1-S 1 . (3.2)
It is recognized that this approach may overestimate the soil deposi-
tion since inedible parts of food crops will intercept depositing
radionuclides. Radionuclides intercepted by inedible portions may all
eventually reach the soil when the field is plowed following harvest,
less any radioactive decay during retention on plant surfaces.
3.2 Time-dependent Root Uptake (Tprt)i
Time dependence of root uptake is especially important when con-
sidering radioactive daughter ingrowth and soil depletion of a particu-
lar radionuclide. It is expected that root uptake will be a function
of the increase in biomass of a plant or plant part over time as well
as of the physiological stage in the life cycle of the plant. Both
biomass increase and physiologic maturation may involve active and
passive processes by which plant tissues incorporate elements. Many
essential elements are both actively and passively acquired, while
other elements may only be passively acquired. Because many radionu-
clides are radioactive isotopes or chemical analogs of essential ele-
ments, root uptake rates should be described in an element-, as well as
time-, dependent sense.
Literature reviewed to date indicates a paucity of data regarding
time-dependent root uptake of most elements. What is available indi-
cates that the shape of the uptake curve, however, is similar to that
19
of the growth curve for some elements and crop species studied.1° - 16
Therefore, the approach adopted in RAGTIME is either to characterize
uptake rates by edible portions of crops on the basis of any empirical
data available, or to assume that the uptake rate follows growth curve
for the edible portion in the absence of empirical uptake data.
The uptake curve obtained via either of these two approaches is
then adjusted such that the concentration in the edible portion for a
chosen harvest time is related to the soil concentration at that time
vy the eirically derived concentraton factor, .. " TheIII valu f oIV
Biv is obtained from empirical studies which measure the final crop
concentration with respect to a soil concentration believed to be
approximately constant throughout the growth cycle. Thus, for elements
whose concentrations are significantly decreased in the root zone
either by movement downward into the soil sink or uptake by crops, con-
centration factors derived from initial soil concentrations and final
crop concentrations may not be appropriate.
3.3 Time-independent Parameters ( e,s' Tsp and Tp,h)
Retention, both initial and long-term, of intercepted radionu-
clides will depend on 8
1. the surface characteristics of the edible portion,
2. the particle size,
3. the wind velocity, and
4. the relative humidity and amount of rainfall.
The effect each of these will have on retention will vary from site to
site, and thus, the value of Te,s' the retention coefficient, may vary
greatly. For the present, we have assumed an average, time-dependent
value of Te,s of 0.0495 day- 1 , consistent with that provided in TER-
MOD, 3 until further research into this parameter can be undertaken.
The movement of radionuclides deposited on surface soil to the
root zone has been characterized in TERMOD as T The definition and
value of this parameter has been carried over to RAGTIME, pending
future investigation into its appropriateness. It is possible that
20
this parameter might best be described in a time-dependent sense, with
empirical data being derived from soil distribution coefficients, or
Kd s, available in the literature. Becaýuse these Kd's are element-
specific, it may also be necessary to incorporate nuclide-specificity
into the definition of Ts,p
The downward movement of radionuclides out of the root zone into
the soil sink is again characterized by a time- and nuclide-independent
parameter, p,h' adopted from TERMOD. As with Ts, further research
Into the appropriateness of the value and interpretation of this param-
eter is pending.
4. PARAMETERS DESCRIBING TRANSFER TO BEEF AND MILK
Contamination of beef and milk may occur as a result of the inter-
ception or root uptake of depositing radionuclides by forage crops, and
the subsequent ingestion by beef or dairy cattle. The RAGTIME code, to
date, considers that this contamination occurs only through grazing of
exposed pasture grasses by cattle. Thus, the loss and/or buildup of
radionuclides present in stored feeds and hay, upon which cattle may
depend for a large portion of the year, are not considered at this
time. Furthermore, inhalation of radioactivity by cattle is not yet
treated explicitly.
4.1 Contamination of Pasture Grass
As with food crops (Sect. 3), pasture grass (Gi) may be contami-
nated through interception of depositing radionuclides, including re-
suspended particulates, and through root uptake of nuclides deposited
on the soil or root mat (Ri) below the pasture grass. At present, a
pasture exposed to depositing radionuclides is assumed to maintain an
approximately constant plant biomass throughout the year, and that the
interception fraction for grasses, S3 , remains constant. The assumed
value of S3 , equal to 0.25, is equivalent to the value originally used
in TERMOD, 3 and falls in the range of empirical measurements reported
by Chamberlain" 7 for initial retention (where sampling is done immedi-
ately after contamination) by grasslands. This parameter would be
expected to vary with plant density and other environmental factors,
and thus represents an average value here. The fraction, S 3 , is
applied directly to the- aerosol source term, Fi (see Fig. 2.1), and
thus the model does not explicitly account for interception of radio-
nuclides resuspended from the soil or root mat below the pasture grass.
The fraction S4 , in Fig. 2.1, represents the fraction of deposit-
ing activity not initially intercepted by grass leaves and thus the
fraction deposited on the surface soil or root mat below the leaves.
Therefore, this value is assigned a constant value of 0.75, being de-
fined as follows:
21
22
S4 1- (4.1)
The value of the parameter -r,g' representing additional input
into the pasture grass compartment from surface soil, is consistent
with the TERMOD value 3 adopted from a paper by Menzel, 7 which indicates
that an upper limit for uptake of radionuclides in the surface soil by
a single crop is 1%. Considered on an annual basis, Trg becomes 2.74
x 10- day-'.
Three aspects of root uptake by ntuiirp grasses have not been con-
sidered at present. First, the element dependency of this parameter,
Trg has been neglected, yet may be quite important when root uptake
is significant with respect. to foliar contamination. Second, an addi-
tional mode of root absorption of radionuclides which does not involve
the soil may be quite significant. This latter mode of uptake involves
the radionuclide availability for uptake from the root mat, which is a
"thatch" of dead and decomposing tissues around the plant-base region
in which grasses may root. 7 Finally, the time dependency of Tr,g has
not been investigated. All of these aspects will be addressed as work
continues on RAGTIME.
Loss of radionuclides from the grass compartment, Gi (see Fig.
2.1), may occur through ingestion of grass by grazing cattle (Tg,),
R g,radioactive decay (X,), and by weathering of surface-deposited radio-
nuclides (g,r ). The value of Tg,r is assumed to be equivalent to the
weathering coefficient, Te,s' discussed in Sect. 3, and thus represents
a 14 day half-time for retention of intercepted materials. This value
is consistent with data reported by Chamberlain" 7 for grasslands,
although it may vary with seasons and climatic factors. In particular,
this weathering coefficient, when measured, will incorporate loss of
surface material due to shedding of the protective leaf cuticle during
plant growth,' 7 thus suggesting a seasonal and species dependency of
Tg~r" As with TeXs' Tg,r is assumed to be time independent until fur-
ther research dictates that a different approach should be taken.
23
The value of -g * (day-) will depend on the rate of loss of
radionuclides in pasture grass through consumption by grazing beef and
dairy cattle. For this model, an average ingestion rate, Vc, of 10 kg
day- 1 dry matter was assumed, consistent with the value used in TER-
MOD. 3 Using this ingestion rate and a dry-weight areal grass density,
Dg, of 0.15 kg m-2 (Ref. 3), we define the value of T, * to be
Vg _ 6.67 x 10-1 day- 1 (4.2)
g - A D Agg g
where
Ag pasture area per cow (m2 ).
At present, it is assumed that 'g, is constant throughout the year.
Further work on incorporating seasonal aspects into the model will
modify this approach.
The rate of loss of radionuclides from the surface soil (Ri) be-
neath pasture grass is represented by the parameter Tr,d* As with the
similarly defined parameter, Tp,h' for crop soil (Sect. 3), an element-
independent rate of 1.096 x 10-4 day-' is used, as given in documenta-
tion of the TERMOD code. 3 Again, further research may indicate a more
appropriate value or representation of this process.
4.2 Contamination of Beef and Milk
Transfer of radionuclides from pasture grass to beef or milk is
parameterized by (T g,b)i, in m2 kg- 1 day-', or (Tgc )i, in m2 liter-1
day-', respectively (see Fig. 2.1). These parameters represent trans-
fer rates and are assumed to be time independent pending further inves-
tigation into data available regarding their time dependency. Element-
specific values of (T)g,b) and (Tgc )i were calculated from empirically
derived transfer coefficients,' 8 ,1 9 (Ff)i and (Fm)i, which characterize
the ratios between beef or milk concentrations of an element and the
equilibrium concentration of that element in pasture grass or feed. By
definition
24
Ff = the fraction of the daily intake of an element by abeef cow which appears per kg of flesh at time ofslaughter (day kg- 1 ), and
F = the fraction of the daily intake of an element by am dairy cow which appears per liter of milk at equilib-
rium (day liter-t).
Therefore, the empirical coefficients represent the theoretical coeffi-
cients only if pasture grass (or feed) is the only source of the ele-
ment in question in the cow's diet. The parameter (T g,b)i was derived
by assuming that the concentration in beef at time of slaughter approx-
imates an equilibrium concentration, given an equilibrium concentration
in pasture grass. Thus, if (B e )i is taken to represent the equilib-cow1bium concentration of an element, i, in the muscle of a single cow
(pCi kg- 1 ), and G q is the equilibrium concentration in grass (pCi
m 2 ), then from the equilibrium equation
d(Bcow)i eq (r e qw)i =
dt = (g,b)iGq - exc i(cow i (4.3)
it follows that
(Beq )i = (Tg'b)i
cow_ _ (4 .4 )GOq (T exci
where
(Texc)i = loss rate of the element, i, from the muscle of a steer
(day-'). Since
(Tg,b) - Gq (exci (4.5)g~b Gýq e1
and
25
(F). = c i(4.6)(G q x V /D)1 cg
where
Vc = dry-weight grass consumption per day by a cow (10 kgday 1),
Dg = dry-weight area] grass density (0.15 kg m-2 ),
it follows that the expression for ( g,b i is related to (Ff)i by:
( (Ff)i(T exc)iVc 47)(g,b)i =D
g
Assuming the animal's diets to consist solely of pasture grass, the
loss rate of the element from the muscle of a steer, (T ex)i, may be
interpreted to represent both the element-specific metabolic turnover,
as well as the element-independent dilution of the concentration due to
increase in muscle mass during growth. This approach has been adopted
from the TERMOD code 3 for the present, but will probably be revised to
reflect a dynamic, rather than steady-state, approach to modeling this
pathway as model development progresses. In doing so, the dilution due
to growth may be handled explicitly rather than incorporated into a
term such as (T ex)i.
Similarly for (tgc )i, representing transfer of element, i, to
milk (C.),
(T ~(F mY(mi lk)Vc 48
('g,c9 i D m g (4.8)
where
Tmilk = the element-independent loss rate from the udder(2 day-').
26
The value of Tmilk in this case corresponds to the frequency of milk-
ing, assumed to be twice daily, and other losses are considered negli-
.gible. Equation (4.8) was derived in a manner similar to Eq. (4.7),
from
d(Ccow)i (T G q e q ). 0
dt g,c ii -milk(cowi (4.9)
and
(C eq )i(F).= cowi (4.10)(m i =(G eq x Vc/Dg
1 cg9
Again, dynamics related to maturation and milking practices for a
single cow have been neglected at this time, but will be considered as
model development progresses.
The equation describing radionuclide concentrations in the beef
compartment as whole [Eq. (2.8)], given in Sect. 2.3 of this report, as
dB. i-1dB T -i i E R +.T i B. + AR i .-
dt (tgb)iGi -i + tbeef + (Texc)i i + - B..Bj=l
differs from that for a single cow given in Eq. (4.3) due to the pres-
ence of an additional element-independent loss parameter, Tbeef* The
interpretation of this aspect of the beef compartment has been adopted
here from TERMOD, 3 in that the compartmental equation considers losses
from beef in the herd as a whole by including the term, Tbeef' to
account for slaughter of contaminated cattle. This interpretation then
implies instantaneous replacement of the slaughtered portion with un-
contaminated cattle and a subsequent reduction or loss of radioactivity
from the compartment. The uncontaminated cattle then begin to accumu-
late radioactivity at a rate determined by (Tg,b)i. If, however, the
27
radionuclide concentration in the beef compartment is to be used as an
indication of man's radiation exposure via ingestion of beef, the
present methodology may underestimate concentrations in beef of cattle
being slaughtered. That is, this latter portion of the herd will like-
ly be the more mature segment which has been exposed to contaminated
pasture for the greatest length of time, although the concentration
calculated will be an average of all members of the herd. In light of
this potential deficiency, work is ongoing to revise the homogeneous
herd concept, where uncontaminated and contaminated beef are indiscrim-
inately mived to produce an average concentration in the beef which may
be lower than that in cattle ready for slaughter.
The milk compartment may also be interpreted to represent concen-
trations in milk obtained from the dairy herd as a whole. In this
case, however, instantaneous replacement of milk removed from the udder
by uncontaminated milk does not result in a reduction in concentration
below that to which man might be exposed, because each lactating cow,
as well as the herd, is subject to this same removal process. That is,
while slaughtering will not affect the radionuclide concentration in
beef of any particular cow, milking will affect the concentration in
milk of each individual lactating cow in the herd, and thus can be con-
sidered when considering the herd as a whole.
For both milk and beef compartments, radionuclide loss and buildup
of daughters due to radioactive decay during storage prior to human
consumption is considered. Compartments (CH)i and (BH)i, representing
concentrations of each nuclide in milk and beef, respectively, follow-
ing storage were devised to provide this information.
5. SOLUTION OF THE SYSTEM OF DIFFERENTIAL EQUATIONS
5.1 Use of the GEAR Subroutine
The system of differential Eqs. (2.1) through (2.11) is solved in
the RAGTIME code by use of the GEAR package 4 for solution of systems of
ordinary, differential equations. The subroutine CALC of RAGTIME makes
a call *to the subroutine GEAR of the GEAR package at each output time
to determine the values of the state variables,
Ei, Si, Pi. Gi. Ri. Di. Ci. B.. TiV H.i and (EH)i
where i = 1 to n (the number of nuclides in the chain) at the given
time. In the notation of the GEAR package, the system of differential
Eqs. (2.1) through (2.11), with i varying from 1 to n, has the form
dY/dT = F(Y,T)
where Y = [Y(1), Y(2), ... , Y(N)] is the state vector at time T, with N
representing the number of state variables. The present version of
RAGTIME uses N = 12 x n state variables (n = the number of nuclides in
the chain). The correspondence between RAGTIME state variable names
and those used in the GEAR package is shown in Table 5.1.
The user of the GEAR package furnishes a subroutine DIFFUN(N,T,Y,
YDOT) which computes the function YDOT = F(Y,T), the right-hand side of
the system of ordinary differential equations, where N, T, and Y are as
described above. The correspondence indicated above between RAGTIME
state variable names and those used by the GEAR package implies a simi-
lar correspondence between the two notations for derivatives (Table
5.2).
The notation used in subroutine CALC of RAGTIME for a call to the
GEAR subroutine for values of the state variables at the time TOUT is
CALL GEAR (DIFFUN, PEDERV, N, TO, HO, YO, TOUT, APS, MF, INDEX)
29
30
Table 5.1 Definition of RAGTIME state variablesin terms of GEAR package notation
RAGTIME GEARstate variable package notation
E. Y[12(i - 1) + 1]1
S. Y[12(i - 1) + 2)1
P. Y[12(i - 1) + 3)1
G. Y[12(i - 1) + 4]
R. Y[12(i - 1) + 5]1
D. Y[12(i - 1) + 6)1
C. Y[12(i - 1) + 7]l
B. Y[12(i - 1) + 8.]1
(EH)i Y[12(i - .1) + 9]
T. Y[12(i - 1) + 10]1
H. Y[12(i - 1) + 11]1
31
Table 5.2 Definition of RAGTIME derivatives interms of GEAR package notation
RAGTIME GEARderivatives package notation
dEi/dt YDOT[12(i - 1) + 1]
dSi/dt YDOT[12(i - 1) + 2]
d.Pi/dt YDOT[12(i - 1) + 3]
dGi/dt YDOT[12(i - 1) + 4]
dRi/dt YDOT[12(i - 1) + 5]
dD.i/dt YDOT[12(i - 1) + 6]
dC4/dt YDOT[12(i - 1) + 7]
dBi/dt YDOT[12(i - 1) + 8]
d(EH)i/dt YDOT[12(i - 1) + 9]
dTi/dt YDOT[12(i - 1) + 10]
dHi/dt YDOT[12(i - 1) + il]
32
where the parameters have the following meanings:
(1) DIFFUN is the name of the subroutine described above,
which is declared external in subroutine CALC.
(2) PEDERV is also a subroutine which is declared external
in CALC. Under certain options available to the user
of the GEAR subroutine, this subroutine is used to de-
fine the N by N Jacobian matrix of partial derivatives.
However, under the option used by RAGTIME, PEDERV is a
dummy subroutine.
(3) N is the number of state variables (i.e., N = 12 x n in
our case).
(4) TO is the initial value of T, the time variable (used
only on the first call).
(5) HO is the step size for T (used only on the first call).
(6) YO is a vector of length N (= 12 x n) containing the
initial values of Y. This vector is used for input
only on the first call.
(7) TOUT is the value of T at which output is desired.
(8) EPS is the relative error bound (used only on the first
call unless INDEX = -1).
(9) MF is a parameter used to indicate the basic method to
be used for integration (Adams method or the stiff
method of GEAR) and the method of iteration.
(10) INDEX is an integer used to indicate the type of call.
Initially, INDEX is set to 1. The value returned for
INDEX is 0 unless the integration was halted for some
reason. For meanings of the values -1, -2, -3, or -4,
for an output value of INDEX, see a listing of the GEAR
package.
33
5.2 Use of the Bateman Equations as a Check
Since RAGTIME uses a numerical method for solution of the system
of differential Eqs. (2.1) through (2.11), it is desirable to have a
procedure for checking the accuracy of this solution. Fortunately, it
is possible to calculate the total amount of radioactivity in the sys-
tem at any given time using an explicit solution of the Bateman equa-
tions. This value can then be compared with the corresponding value
calculated by summing the amounts of radioactivity in the various com-
partments of the model as calculated by use of the GEARK package. Close
agreement of these two values is a necessary but not sufficient condi-
tion that the numerical solution of the model equations is accurate to
the degree desired. In particular, one should keep in mind that a
large relative error in a compartment whose radioactivity contribution
is small compared to other compartments would be masked by this summing
procedure. Nevertheless, the comparison of total radioactivity as cal-
culated in these two ways provides valuable assistance in evaluating
the numerical method since, as most practitioners of numerical analysis
would admit, the use of such methods is still largely an empirical
science. In particular, this comparison can provide guidance for the
selection of appropriate options and parameter values to be used in
calls to the GEAR subroutine.
The Bateman equations describe the decay process of a radionuclide
chain. Consider a chain of radionuclide species indexed i = 1, ... , n
in a comparment into which the exogenous inflow rate of the ith species
is given by li(t)(pCi day-') and which is subject to first-order re-
moval processes with removal constant, XB (day-'). Then the following1
system of differential equations describes the decay process in this
compartment:
dA. i-1
(X + X )Ai + X B1A + I(t), i =1, ... , n (5.1)1 11 1j=l
34
where
Ai(t) = radioactivity (pCi) of the ith nuclide at time t(day),
= decay constant (day-') for the ith nuclide,1
X B, = removal constant (day-') for the ith nuclide,
B.. = radioactive branching ratio from species j toli species i, j < i,
li(t) = exogenous inflow rate for species i (pCi day-').
For the purpose of checking the total amount of activity in the RAGTIME
compartments against the value as predicted using the Bateman equa-
tions, we may regard the total exogenous inflow rate for species i into
the system to be the product of the deposition source F.(t) (pCi m-2
day- 1 ) and the quantity
SiA + S2 A + S3 A + S4Ag = (S, + S 2 )A + (S3 + S4 )Ag , (5.2)
where
A = soil surface area (m2 ) assumed for the above-surfacefood crop,
A = soil surface area (m2 ) assumed for the pasture grassg compartment.
Subroutine CHECK of RAGTIME makes a call to subroutine TRAFUN for the
purpose of calculation of the total radioactivity in the system at
various times. TRAFUN requires that the function representing the
exogenous input rate be a step function of time, which dictates that
Fi(t) be a step function. Furthermore, TRAFUN requires that the exoge-
nous input rate be prescribed as a doubly dimensioned array rather than
as a FORTRAN function. We thus reserve for Sect. 6.3 an explicit
description of the call by CHECK to TRAFUN, in order to make use of our
description in Sect. 6.1 of the doubly dimensioned inflow rate matrix
35
FF which is defined in subroutine INPUT and whose values are used to
define both the deposition source function Fi(t) [in FORTRAN, F(I,T)]
and the doubly dimensioned exogenous input rate matrix P which is used
as an input parameter to TRAFUN.
6. THE RAGTIME CODE
6.1 Input
The subroutine INPUT reads values for user-supplied data required
by RAGTIME. The first value read is that for NUMNUC, the number of
nuclides in the chain. Next read are the names of the nuclides
[NAMNUC(I), I = 1 to NUMNUC] and the initial ground deposition source
(microcuries per square meter) for each nuclide [FO(I), I = 1 to NUM-
NUC]. Following these steps values are read for MP and for the arrays
TIMEP(KP), KP = 1 to MP
and
FF(I,KP), I = 1 to NUMNUC, KP 1 to MP
where
FF(I,KP) inflow rate of species I (microcuries m-2 day-)for the time interval TIMEP(KP) to TIMEP(KP+1)if KP < MP,
FF(I,MP) inflow rate at times subsequent to TIMEP(MP).
The matrix FF defines the exogenous input of radioactivity into the
system. This matrix is used to define values of the fallout source
function Fi(t) [in FORTRAN, FUNCTION F(I,T)]. Also FF is used in sub-
routine CHECK to define the exogenous input rate matrix P as discussed
in Sects. 5.2 and 6.3. Next read are the number (NUMBRA) of and values
for the radioactive branching ratios. The FORTRAN notation B(.I,J) is
used to denote the radioactive branching ratio from species J to spe-
cies l(J < I). If NUMBRA = 0, no branching ratios are read. If NUMBRA
> 1, then one card is read for each nonzero branching ratio, the READ
statement and its associated FORMAT being
READ(RDR,50) PARNUC,DAUNUC,BRATIO
50 FORMAT(A8,5X,A8,5X,E13.6)
37
38
where PARNUC is the name of the "parent" and DAUNUC the name of the
"daughter." For example, the card
B1210 P0210 1.0t
col. 1 col. 14 cols. 27-40
is used to input the branching ratio 1.0 from 21 0 Bi to 2 1 °Po. The in-
put subroutine assigns the value BRATIO to the element B(I,J) of the
matrix of branching ratios in such a manner that B(I,J) represents the
branching ratio from the nuclide NAMNUC(J) to NAMNUC(I).
Following input of the branching ratios, information regarding
desired output times is read. The entire integration interval is spec-
ified as consisting of a number (NINTVL) of subintervals, with INCR(I)
denoting the interval between successive output times for the subinter-.
val indexed by I. The right endpoint of the subinterval indexed by I
is denoted by ENDTIM(I). The READ statement and its associated FORMAT
statement for input of these quantities is
READ(RDR,90) INCR(I),ENDTIM(I)
90 FORMAT(E13.6,16)
In subroutine CALCIN, the arrays INCR and ENDTIM are used to define
values for the entire array of output times TIM(I), I = 1 to NTIM.
Values for the radionuclide-dependent parameters listed in Table
6.1 are read next. For definitions of these parameters, see Table 2.1.
One card is read for each of these parameters, the cards being read in
the order indicated in Table 6.1 for each value of I, with I varying
from 1 to NUMNUC. The format for each card is (1OX,E13.7).
We now consider the input of nuclide-independent parameters. The
main program of RAGTIME handles any number of radionuclide chains. The
first executable statement of MAIN is
READ(RDR,10) NCHAIN
10 FORMAT(1OX,12)
39
Table 6.1 Radionuclide-dependent inputparameters for RAGTIME
Parameter symbol FORTRAN name
x R LAMRR(I)
(Fm)i FSUBM(I)
(Ff)i FSUBF(I)
B. BSUBIV(I)
(Texc)i TAUEXC(I)
where NCHAIN is the number of chains to be considered. Following this
read statement, the main program makes calls to the subroutine INPUT,
OUTDAT, CALCIN, HARVST, CALC, and CHECK in this order for each chain
under consideration. In order to avoid the necessity of inputting the
nuclide-independent parameters for each chain, a flag, IFLAG, is set in
MAIN and passed as a parameter in the call to INPUT [CALL INPUT(IFLAG)]
to enable a branch to be made around the input of these parameters for
calls subsequent to the first one. Values are read for the nuclide-
independent parameters listed in Table 6.2. For definitions of these
parameters, see Table 2.1. One card is read for each of these param-
eters, the format being (10X,E13.7).
The final segment of code in INPUT reads values for certain GEAR
subroutine parameters (see Sect. 5.1).
6.2 Logical Structure of the Code
After calling subroutine INPUT for input of data related to a
given chain, the main program calls OUTDAT for a printout of these
data. Next a call is made to subroutine CALCIN, which calculates the
values of certain coefficients used in the system of differential Eqs.
(2.1) through (2.11). These coefficients, the numbers of the equations
in which they occur, and their FORTRAN designations are as follows:
40
Table 6.2 Nuclide-independent inputparameters for RAGTIME
Parameter symbol FORTRAN name
A
Ag
d
Dg
Mb
P
T beef
T milk
T
g,r
p,h
r,d
r,gT
5s,p
U
VC
htb
thC
A
ASUBG
SMALLD
DSUBG
MSUBB
RHO
TAUBEF
TAUMLK
TAUES
TAUGR
TAUPH
TAURD
TAURG
TAUSP
U
VSUBC
TIMBH
TIMCH
41
Occurs in FORTRANCoefficient equation name
A+ + Te 2.1 LAMA(ID1 ,
xR + T 2.2 LAMS(I)
x +g• + T * 2.4 LAMG(I)l g,r g,
R+i + Tmilk 2.7 LAMC(I)
Subroutine CALCIN also defines values of the array of output times
TIM(I), I = 1 to NTIM as discussed in Sect. 6.1. Following the call to
CALCIN, the main program calls subroutine HARVST in which emergence and
harvest times for crop plants are specified. The FORTRAN name for the
number of harvests considered is NHARV. Subroutine HARVST assigns
values to NHARV and to the arrays EMERGE(I), I = 1 to NHARV, and HAR-
TIM(I), I = 1 to NHARV, where
EMERGE(I) date (days) for Ith emergence time for plants,
HARTIM(I) = date (days) for Ith harvest time for plants.
After calling HARVST, MAIN calls subroutine CALC which serves the
following functions [(1)-(4)]:
(1) Initializes and prints definitions of compartments
Ei' Si, Pi' Gi' Ri. Ci' Bi, Ti, (EH)., (BH)i, and (CH)i.
(2) Assigns values to GEAR subroutine parameters INDEX, TO,
HO, and N (Sect. 5.1).
(3) Assigns values to the arrays TR(INUC), TB(INUC), and
PRORAT (INUC), INUC = 1 to NUMNUC, where
42
TR(INUC) radioactive half-life (days) ofnuclide INUC,
TB(INUC) = biological half-time (days) ofnuclide INUC (a large value isassigned to approximate a biolog-ical removal factor of zero),
PRORAT(INUC) = (constant) production rate fornuclide INUC.
These arrays are used as input parameters to the sub-
routine RESDNS, which is called by CALC for calculation
(using an explicit solution of the Bateman equations) of
the radioactivity level in the holdup compartments (BH)i
and (CH)i as discussed in Sect. 2.2. The array PRORAT
is used to specify the production rate in compartment
(BH)i and is therefore assigned zero values since no
exogenous input is assumed for this compartment.
(4) Following steps (1) through (3), subroutine CALC executes
a loop,
DO 4 1 = 1,NTIM
in which the values of the state variables defined by
Eqs. (2.1) through (2.11) as well as the holdup compart-
ments (BH)i and (CH)i are computed and printed. At each
time TOUT = TIM(I), a call is made to the GEAR subrou-
tine,
CALL(DIFFUN,PEDERV,N,TO,HO,YO,TOUT,EPS,MF,INDEX)
for calculation of values of the state variables Ei, Si,
Pi' Gi' Ri' Di' Ci' Bi, Ti' Hi and (EH)i, where i 1 to
the number (NUMNUC) of nuclides in the chain. These
values are returned in the array YO, with the same
43
correspondence between RAGTIME state variable names and
GEAR package names as indicated in Sect. 5.1;-for exam-
ple, the value of E. is given by YO[12(i - I1 N 1].
At each output time, CALC calculates and saves in the array TOTUCI
the total activity in the system corresponding to each nuclide. We
define
TOTUCI(i,!NUC) total activity (microcuries) in the system
corresponding to nuclide INUC at time
TIM(I).
If we assume A square meters of land devoted to crop production and Agsquare meters of pasture grass, the appropriate multiplicative factors
for converting the values in the various compartments to microcuries
are given in Table 6.3. The values of these conversion factors are
obvious for all except compartments Ci, Bi, and (EH)i. For the milk
compartment Ci, we assume one cow for each A square meters of pasture
grass, with an udder capacity of U liters per cow. Similarly, the con-
version factor for the beef compartment follows from our assumption of
Mb kilograms of muscle per steer, with one steer per A square meters
of pasture grass. The conversion factor M /1000 for the crop holdup
compartment (EH)i is based on the definition
M0 = total mass (grams) of crop per A m2 at harvest time.f
The compartment MW' is used only in connection with a mass-balance
check. This compartment serves to account for the loss to the system
of activity from the beef and milk compartments Bi and Ci as well as
the complement of the activity represented by transfer coefficients
(Tg,b)i and (TgC )i with respect to T ,. The differential equation
for M' is1
44
Table 6.3 Conversion factors used to convert compartmental
concentrations to total activity
Compartment
E.1
S.
P.
G1*
R
D
C.
B.
(EH)i
T.1
H.M
M.I
Units
pCi m-2
pCi m-2
pCi
pCi m-2
pCi m-2
pCi m-2
pCi liter-'
pCi kg-I
pCi kg-I
pCi
pCi m-2
pCi
FORTRAN name
YO[12(i - 1) + 1]
YO[12(i - 1) + 2]
YO[12(i - 1) + 3]
YO[12(i - 1) + 4]
YO[12(i - 1) + 5]
YO[12(i - 1) + 6]
YO[12(i - 1) + 7]
YO[12(i - 1) + 8]
YO[12(i - 1) + 9]
YO[12(i - 1) + 10]
YO[12(i - 1) + 11]
YO[12(i - 1) + 12]
Conversion factor
A
A
1
A (ASUBG)Ag(ASUBG)
A (ASUBG)
U
Mb(MSUBB)
M0/1,O00
1
A
1
45
*dM'.RT M +B ± UC. - XRMt + V/D - (T•gbMbdt beef b i milk 1 1 1 + 'gD10
- (Tg c)iUGi, (6.1)
where
Mb mass of muscle on a steer at time of slaughter (kg),
U milk capacity of the udder (liters).
,,e other parameters were defined and discussed in Sects. 2, 3, and 4
of this report. The first two terms on the right-hand side of this
equation represent gains to M! resulting from losses to compartments B.
and Ci through slaughter of cattle and milking of cows, respectively.
The third term accounts for radioactive decay in Mi. The last term
represents the loss to compartments Gi which is not accounted for bythe terms (Tg,b)iGi and (T g1).Gi in Eqs. (2.7) and (2.8), respective-
ly. 0With MFO as the FORTRAN name for Mf, the FORTRAN statement defin-
ing TOTUCI(I,1) is therefore
TOTUCI(I,1) = YO(1)*A + YO(2)*A + YO(3) + YO(4)*ASUBG
1 + YO(5)*ASUBG + YO(6)*ASUBG + YO(7)M*U + YO(8)*MSUBB
2 + YO(9)*(MFO/1000.) + YO(1O) + YO(11)*A + YO(13),
(6.2)
the generalization for TOTUCI(I,INUC) being obvious.
After defining values of the array TOTUCI(I,INUC) for a given I
and for INUC = 1 to NUMNUC, calls are made to subroutine RESDNS for the
calculation of values of the beef and milk holdup compartments (BH)i
and (CH). The statements executing these calls are
CALL RESDNS(TIMBH,NUMNUC,TR,TB,B,PRORAT,QOBEF,QBEF,QWIGL,IDIM)
CALL RESDNS(TIMCH,NUMNUC,TR,TB,B,PRORAT,QOMLK,QMLK,QWIGL,IDIM)
46
where TIMBH, TIMCH, NUMNUC, TR, TB, B (matrix of branching ratios) and
PRORAT are as defined previously. On input, the arrays QOBEF and QOMLK
contain the values in the beef and milk compartments Bi and Ci, respec-
tively, at the current time TIM(I). The concentration levels (pCi kg- 1 )
which beef and milk would reach if stored for the period of time (days)
specified by TIMBH and TIMCH are returned in the arrays QBEF and QMLK,
respectively. The array QWIGL contains residence values. (Ci-days) on
output and is not used by RAGTIME. The parameter IDIM specifies the
maximum dimensions for the matrix of branching ratios as defined in
RESDNS (REAL BRANCH(IDIM,IDIM)). Following these calls to RESDNS,
values of the various compartments, including the holdup compartments
for beef andmilk, are printed for the current time.The final section of code within the I-loop determines whether or
not the current time, TIM(I), is a harvest time, and if so, reini-
tializes the state variables Ei, Ti, and (EH)i to simulate harvest.
First the following call is made to subroutine QUERY:
CALL QUERY(TIM(I),IANS)
This subroutine searches the array of harvest times, HARTIM, and
returns IANS = 1 if TIM(I) is a harvest time, IANS = 0 if not. If
TIM(1) is a harvest time, the crop holdup compartment (EH)i is reini-
tialized to the value
total activity (pCi) in compartments Ei and Ti AEi + Ti
total mass (kg) of crop at harvest time Mf x 0.001
(6.3)
after which the compartments Ei and Ti are set to zero. After comple-
tion of the I-loop, control returns to the main program.
As pointed out in Sect. 5.1, the system of differential Eqs. (2.1)
through (2.11) is defined in subroutine DIFFUN. The GEAR package makes
calls to DIFFUN for values of YDOT at various times as described in
Sect. 5.1. The deposition source function Fi(t) of Eqs.. (2.1), (2.2),
47
(2.4), and (2.5) is defined by means of the FORTRAN function F(I,T).
This function is defined in terms of the inflow rate matrix
FF(I,KP), I = 1 to NUMNUC, KP = 1 to MP
as. described in Sect. 6.1..
The final subroutine called by MAIN is subroutine CHECK. Details
concerning this subroutine are discussed in the following section.
6.3 Subroutine CHECK
The last subroutine called by MAIN is subroutine'CHECK. This sub-
routine calculates and prints values of
TOTUCI(ITIM,INUC) =
ACT(.INUC,ITIM)
total activity ([Ci) due to nuclide INUC
at time TIM(ITIM) as calculated in sub-
routine CALC by calls to the GEAR sub-
routine (discussed in Sect. 6.2)
total activity (pCi) due to nuclide INUC
at time TIM(ITIM) as calculated by sub-
routine TRAFUN, using the Bateman equa-
tions,
and the percentage error
(T0TUCI(ITIM,INUC)) - ACT(INUC,ITIM)*100./ACT(INUC,ITIM)
for times TIM(ITIM), ITIM = 1 to NTIM. The call to TRAFUN is as fol-
lows:
CALL TRAFUN(NUMNUC,TR,TB,B,MP,TIMEP,P,NTIM,RTIM,AWIGL,ACT,IDIM)
The meanings of NUMNUC, TR (radioactive half-lives), TB (biological
half-times), B (matrix of branching ratios), MP (Sect. 6.1), TIMEP
(Sect. 6.1), NTIM (number of output times), and IDIM are as defined
48
previously. The array RTIM is a real array whose values are the same
as those for the double precision array TIM of output times. The
matrix
P(INUC,KP), INUC = 1 to NUMNUC, KP = 1 to MP
defines the total exogenous input rate of each nuclide to the system
for each of the time intervals TIMEP(KP), KP = 1 to MP. From the
definition of the inflow rate matrix FF (Sect. 6.1) and our assumption
of A square meters of crop production land and A square meters ofgpasture land, it follows that the correct expression for P(INUC,KP) is
P(INUC,KP) = FF(INUC,KP)*((S1 + S2)*A + (S3 + S4)*ASUBG)
Subroutine TRAFUN returns in the array ACT the total activity levels as
computed using the Bateman equations. After the call to TRAFUN, the
values of the arrays TOTUCI and ACT as well as the percentage errors
are printed.
ACKNOWLEDGEMENT
The authors wish to acknowledge the valuable assistance of Ms.
Angela J. Moorman and Mr. Rudolph Kuehn, who were at the time of their
involvement with this project, students of computer science and mathe-
matics at East Tennessee State University.
49
REFERENCES
1. F. 0. Hoffman, C. W. Miller, D. L. Shaeffer, C. T. Garten, Jr.,R. W. Thor, and J. T. Ensminger, A CompZZilation of Documented Com-puter Codes Applicable to Environmental Assessment of RadioactivityReleases, ORNL/TM-5830, Oak Ridge National Laboratory, Oak Ridge,Tennessee (April 1977).
2. U. S. Nuclear Regulatory Commission, Calculation of Annual Dosesto Man from Routine Releases of Reactor Effluents for the Purposeof Evaluating Compl,,!iance with 10 CFR Part 50, Appendix I, Regula-tory Guide 1.109, Revision 1, USNRC, Washington, D. C. (October1977).
3. R. S. Booth and S. V. Kaye, A Preliminary Systems Analysis Modelof Radioactivity Transfer to Man from Deposition in a TerrestrialEnvironment, ORNL/TM-3135, Oak Ridge National Laboratory, OakRidge, Tennessee (October 1971).
4. A. C. Hindmarsh, GEAR: Ordinary Differential Equation SystemSolver, UCID-30001, Revision 3, Lawrence Livermore Laboratory,Livermore, California'(December 1974).
5. J. P. Witherspoon and F. G. Taylor, Jr., "Interception and Reten-tion of a Simulated Fallout by Agricultural Plants," Health Phys.19, 493-499 (1970).
6. J. D. Zimbrick and P. G. Voilleque, ed., 1967 CERT Progress Re-port, Controlled Environmental Radioiodine Tests at the NationalReactor Testing Station, Progress Report Number Four, IDO-12065,Idaho Falls, Idaho (December 1968).
7. R. G. Menzel, "Factors Influencing the Biological Availabilityof Radionuclides for Plants," Fed. Am. Soc. Exptl. Biol. Proc. 22,1398-1401 (1963).
8. C. F. Miller and Hong Lee, Operation Ceniza-Arena: The Retentionof Fallout Particles from Volcan Irazu (Costa Rica) by Plants andPeople, Part One, SRI-MU-4890, Stanford Research Institute, MenloPark, California (January 1966).
9. C. F. Miller, Operation Ceniza-Arena: Retention of Fallout Parti-cles from Volcan Irazu (Costa Rica) by Plants and People, PartThree, SRI-MU-4890, Stanford Research Institute, Menlo Park,California (December 1967).
10. C. R. Belcher and J. L. Ragland, "Phosphorus Absorption by Sod-Planted Corn (Zea mays L.) from Surface-Applied Phosphorus,"Agronomy J. 64, 754-756 (1972).
50
REFERENCES (cont'd)
11. J. L. Brewster, K. K. S. Bhat, and P. H. Nye, "The Possibilityof Predicting Solute Uptake and Plant Growth Response from inde-pendently Measured Soil and Plant Characteristics," Plant andSoil 42, 197-226 (1975).
12. G. L. Pacques, R. L. Vanderlip, and D. A. Whitney, "Growth andNutrient Accumulation and Distribution in Grain Sorghum. I. DryMatter Production and Ca and Mg Uptake and Distribution," Agro-nomy j. 67, 607-616 (1975).
13. L. H. P. Jones and K. A. Handreck, "Studies of Silica in the OatPlant," Plant and Soil 23, 79-96 (1977).
14. J. E. Miller, J. J. Hassett, and D. E. Koeppe, "Interactions ofLead and Cadmium on Metal Uptake and Growth of Corn Plants,"J. Environ. Qual. 6, 18-20 (1977).
15. M. B. Page, J. L. Smalley, and 0. Talibudeen, "The Growth andNutrient Uptake of Winter Wheat," Plant and Soil 49, 149-160(1977).
16. C. L. Parks, A. W. White and F. C. Boswell, "Effect of a PlasticBarrier Under the Nitrate Band on Nitrogen Uptake by Plants,"Agronomy J. 62,437-439 (1970).
17. A. C. Chamberlain, "Interception and Retention of RadioactiveAerosols by Vegetation," Atmos. Environ. 4, 57-78 (1970).
18. R. E. Moore, C. F. Baes III, L. M. McDowell-Boyer, A. P. Watson,F. 0. Hoffman, J. C. Pleasant, and C. W. Miller, AIRDOS-EPA: AComputerized Methodology for Estimating Environmental Concentra-tions and Dose to Man from Airborne Releases of Radionuclides,ORNL-5532, Oak Ridge National Laboratory, Oak Ridge, Tennessee(June 1979).
19. Y. C. Ng, C. S. Colsher, D. J. Quinn, and S. E. Thompson, Trans-fer Coefficients for the Prediction of the Dose to Man via theForage-Cow-Milk Pathway from Radionuclides Released to the Bio-sphere, UCRL-51939, Lawrence Livermore Laboratory, Livermore,California (July 1977).
APPENDIX A
LISTING OF RAGTIME SOURCE CODE
AND
JOB CONTROL LANGUAGE
53
LEVEL 21.8 [ JUN 74 ) OS/360 FORTRAN H
COMPILER OPTIONS - NAME= MAINOPT=02,LINECNT=60,SIZE=OOOOK,SOURCE,BF3COIC,NOLISTNODECKLOADMAP,NOEDIT, ID,NOXREF
ISN 0002 SUBROUTINE INPUT (IFLAGICCC SUBROUTINE INPUT INPUTS ALL THE DATACC
ISN 0003 COMMON /HOLTIM/ TIMBH,TIMCHISN 0004 COMMON /INFLOW/ MP, TIMEP(30), FF(15,30)ISN 0005 COMMON /BRANCH/ 8(15,151ISN 0006 COMMON /DEP/ LAMRPRIS5,FSUeM(15),TAUEXC(15),FSUeFC15),BSUBIV(15)ISN 0007 COMMON /INDEP/ A,ASUBG,DSUBG,MSUBB,TAUBEF,TAUMLK, TAUES,TAUGR,
* TAUPHTAURD,TAURG,TAUSP,UVSUBC,SMALLD,RHO
ISN 0008 COMMON /NAMES/ NAMNUC(15)
ISN 0009 COMMON /NUMBRS! NUMNUCISN 0010 COMMON /TIME/ TIM(365),INCR(30OENDTIMC30),NINTVL,NTIMISN 0011 COMMON ISOURCE/ FO(15)ISN 0012 COMMON /GPARAM/ EPS,MFISN 0013 COMMON /IODEV / PTR,RDR
CCC
ISN 0014 DOUBLE PRECISION NAMNUC,PARNUC,DAUNUC,TIM,EPSISN 0015 INTEGER ENDTIM,PTR,RORISN 0016 REAL MSUBB,INCRvLAMRR
CCCC***
C*** NU14NUC - THE NUMBER OF NUCLIDES IN THE CHAIN BEING STUDIED
ISN 0017 READ (RCR,10) NUMNUCISN 0018 10 FORMAT (131
C*** NAMNUC - ARRAY OF THE NAMES OF 'THE NUCLIDES IN THE CHAIN.C*** FO - INITIAL GROUND DEPOSITION SOURCE FOR NUCLIDE(I)C*** WHERE I VARIES FROM 1 TO THE NUMBER OF NUCLIDESC*** (MICROCURIES PER SQUARE METER)
ISN 0019 DO 30 1 = INUMNUCISN 0020 READ (RDR,20) NAMNUC(IhFO(I)ISN 0021 20 FORMAT (AB,E13.6)ISN 0022 30 CONTINUE
CC*** INFLOW RATES FOR VARIOUS SPECIES
C*** READ VALUES FOR THE ARRAYSC***
C*** TIMEP(KP), KP=l TO MP, ANDC*** FF(ItKPO, I=1 TO NUMNUC, KP=l TO MP, WHEREC***
C*** FF(I,KP) = INFLOW RATE OF SPECIES I (UCI/M'0*2-DAY) FROMC*** TIMEP(KP) TO TIMEP(KP*1) IF KP .LT. MPC_*** AND FF(I,MPl IS THE RATE AT TIMES SUBSEQUENT TOC*** TIMEPIMP).
ISN 0023 READ 4ROR,2001 MPISN 0024 200 FORMAT{15)
54
ISN 0025 DO 201 I=1,NUMNUCTSN 0026 00 201 KP=lMP
ISN 0027 201 READ (RCR,202) TIMEP(KPl,FF( I,vP)ISN 0028 202 FORMAT(E13.6,2X, E13.6)
CCC MATRIX CONTAINING BRANCHING RATIOS INITIALIZED TO ZEROC
C
C*** B - MATRIX COrNTAINING THE BRANCHING RAT!PS.C*-=*
ISN 0029 D0 40 1 = 1,NUMNUCISN 0030 DO 40 J = 1,NUMNUCISN 0031 R(I,Ji = 0.0ISN 0032 40 CONTINUE
CC
C BPANCHING RATIOS FILLED IN MATRIX
CCC***
C*** PARNUC - DARENT NUCLIDEC*** DAUNUC - DAUGHTER NUCLIDEC*** BRATIO - BRANCHING RATIOC*** PARNOD - INDEX INTO MATRIX BC*'* REPPESENTS PARENT NUCLIDEC*** DAtJNOD - INDEX INTO WATRIX B_*** REPRESENTS DAUGHTER NUCLIDEC*** NUMBRA - NUMBER OF NONZERO BRANCHING PATIOSC***
TSN 0033 READ (RCR,10I NUMBRA1SN 0034 IF (NUMBRA.EQ.0I GO TO 160ISN 0036 DO 70 1 = 1,NUMBRAISN 0037 READ (RDR,.50) PARNUC,DAUNUCB3RATIOISN 0038 50 FORMAT iA8,5X,A8,5X,El5.6)
CCC LOOP TO SET APPROPRIATE INDICES INTO MATRIX B
CC
ISN 0039 00 60 J = I,NUMNUCISN 0040 IFIPARNUC .EQ. NAMNUCMJ)) PARNOD = JISN 0042 IF(DAUNUC .EO. NAMNUC(J)) DAUNOD = JISN 0044 60 CONTINUEISN 0045 B(DAUN0D,PARNOD= BRATIOI N 0046 70 CONTINUEISN 0047 80 CONTINUE
C'***
C*** NINTVL - NUMBER OF INTERVALS SPECIFYING OUTPUT TIMESC*** INCRUl) - STEPSIZE FOR INTERVAL IC*** ENDTIM(IM - RIGHT ENDPOINT OF INTERVAL I (DAY)
C*** NOTE THAT ENDTIM(l) SHOULD BE AN INTEGRAL MULTIPLE OF INCR,()C*** AND FOR I GREATER. THAN 1, ENDTIM(II - ENDTIM(I-IP SHOULD BE ANC*** INTEGRAL MULTIPLE OF INCR(I).
55
c*' * *[SN 0048 1(-0 READ {RDR,10) NINTVLISM 0 049 DO 100 I=1,NINTVL!TN 0050 READ (ROR,90) INCR({ ),ENDTIM( I)ISN', D051 qO FORNAT (Elý). o,!6
ISN 0052 100 CONTINUE[SN 0353 110 CONTINUE
C.C
C*** PADIONUCLIDE DEPENDENT PARAMETERS
C* •*C**I BSUBIV - CONCENTRATION OF NUCLIDE PER UNIT FRESH WFIGHT IN PLANTC*** (MICROCURIES PER KILOGRAM) DIVIDED BY CONCENTRATION Dc
*z** NUrCLIDE PER UNIT DRY WEIGHT IN SOIL (MICROCUPIES PERC*v* KILOGRAM)
C~** FSUBF - THE FRACTION OF THE ANIMAL'S DAILY INTAKE OF NUCLIDE(IWHICH APPFAPS IN EACH KILOGRAM OF FLESH (DAYS PER KILOGRAM)
C*** FSURM - FRACTION OF THE DAILY INTAKE OF NUCLIDE I BY A COWC**-*, WHICH APPEARS DER, LITER OF MILK AT EQUILIBRIUMC*** (DAYS PER LITER).C*** LAMRR - RADIOACTIVE DECAY PATE OF THE NUCLIDE UNDER STUDYC*** (PER DAY)C*** TAUEXC - EXCRETION PATE OF A STABLE ISOTOPE OF THE NUCLIDE FROMC*** THE MUSCLE OF A STEER (PER DAY)
C
CC
ISN 0054 00 130 1 = 1,NUMNUCISNI 0055 READ (RDR,1201 LAPRR(I)ISM 0056 READ (RDR,120) FSUBM(1)ISN 0057 READ (RDR,120) FSUBFMI)ISN 0058 READ (RDR,1201 BSU6IV(IlISN 0059 READ (RDR,120) TAUEXC(I)ISN 0060 120 FORMAT (IOXEI3.7)ISM 0061 130 CONTINUE
CC
ISN 0062 IF (IFLAG .FQ. I) GO TO 140ISN 0064 IFLAG : I
CC
C**5*
C*** RADIONUCLIDE INDEPENDENT PARAMETERS
C**5*
C***
C*** A - SOIL SURFACE AREA REQUIRED TO FURNISH FOOD CROPS FOR ONEC*** MAN (SQUARE METERS)C*** ASUBG - PASTURE AREA PER COW (SQUARE METERS)
56
C**4 SMALLD - DEPTH OF THE PLOW LAYER (CENTIMETERS$.C*** RHO - DENSITY OF THE SOIL (GRAMS PER CUBIC CENTIMETER).C*** MSUBB - MASS OF MUSCLE ON A STEER AT THE TIME OF SLAUGHTERC*** (KILOGRAMS PER STEER)C*** TAUBEF - FRACTION OF THE BEEF HERD SLAUGHTERED (PER DAY)C*** TAUMLK - TRANSFER RATE OF MILK FROM THE UDDER (PER DAY)C*** TAUES - TRANSFER COEFFICIENT FROM E TO S (PER DAY)C*** TAUGR - TRANSFER COEFFICIENT FROM G TO R (PER DAY)C** TAUPH - TRANSFER COEFFICIENT FROM P TO H (PER DAY)C*** TAURD - TRANSFER COEFFICIENT FROM R TO D (PER DAY)C*** TAURG - TRANSFER COEFFICIENT FROM R TO G (PER DAY)C*** TAUSP - TRANSFER COEFFICIENT FROM S TO P (PER DAY)C*** U - MILK CAPACITY OF THE UDDER (LITERS)C*** VSUBC - DRY WEIGHT GRASS CONSUMPTION PER DAY-BY A COWC*** (KILOGRAMS PER DAY)C*** TIMBH - HOLDUP TIME (DAYS) FOR BEEFC*** TIMCH - HOLDUP TIME (DAYS) FOR MILK
C
CC
ISN 0065 READ (RDR,001) AISN 0066 READ (RDR,001) ASUBGISN 0067 READ (RCR,001) SMALLDISN 0068 READ (RCR,O011 RHOISN 0069 READ (RDR,O01) DSUBGISN 0070 READ (RDR,001) MSUB9ISN 0071 READ (RM,*001 TAUBEFISN 0072 READ (RCR,00i) TAUMLKISN 0073 READ (RCR,0011 TAUESISN 0074 READ (RDR,0011 TAUGRISN 0075 READ (ROR,O01) TAUPHISN 0076 READ (RCPO01) TAURDISN 0077 READ (RCR,001, TAURGISN 0078 READ (RDR,O01) TAUSPISN 0079 READ (ROR,001) UISN 0080 READ (RDR,001) VSUBCISN 0081 READ (RDR,001) TIMBHISN 0082 READ (ROR,00I) TIMCH
CISN 0083 001 FORMAT (IOXvE13.7)
C
C***
C*** GEAR SUBROUTINE PARAMETERSC*** ***
CISN 0084 READ (RDR,002| EPSISN 0085 002 FORMAT (iOX,D!3.61
CISN 0086 READ (RDR,003) MFISN 0087 003 FORMAT (IOX,12)
CISN 0088 140 CONTINUEISN 0089 RETURNISN 0090 END
57
LEVEL 21.8 ( JUN 74 1 OS/360 FORTRAN H
COMPILER OPTIONS - NAME= MAINOPT=02,LINECNT=6OSIZE=OOOOKvSOURCEEBCDIC,NOLIST,NODECKLOADMAPNOEDITIDNOXREF
ISN 0002 SUBROUTINE OUTDAT
CCC SUBROUTINE OUTOAT ECHOS THE DATA INPUT IN SUBROUTINE INPUTCC
ISN 0003 COMMON /INFLOW/ MP,TIMEP(30), FF(15,30OISN 0004 COMMON /HOLTIM/ TIMBH,TIMCHISN 0005 COMMON /BRANCH/ B(15tI5)ISN 0006 COMMON /DEPI LAMRR(151,FSUBM(15),TAUEXC(15),FSUBF(15),BSUBIV(15)ISN 0007 COMMON /INDEP/ AASUBGDSUBG,MSUBB,TAUBEF,TAUMLK,TAUES,TAUGR,
* TAUPH,TAURD,TAUJRG,TAUSPU,VSUBC,SMALLO,RHOISN 0008 COMMON /NAMES/ NAMNUC(15)ISN 0009 COMMON /NU-,_1!R" S If NUNMNUCISN 0010 COMMON /TIME/ TIM(35),INCR(30),ENDTIM(30),N!NTVLNTIMISM 0011 COMMON /SOURCE/ FO(15)ISN 0012 COMMON /GPARAM/ EPS,MFISN 0013 COMMON /IOEV I PTR,ROR
C
C
ISM 0014 DOUBLE PRECISION NAMNUC,PARNUC,DAUNUCTIMEPSISN 0015 INTEGER ENDTIM,PTR,RDRISN 0016 REAL MSUBB,INCR,LAMRR
CCCC NUCLIDE DEPENDENT PARAMETERS
CC
ISN 0017 WRITE (PTR,40)ISN 0018 40 FORMAT (I'',53X,'NUCLIDE DEPENDENT PARAMETERS' //I
* ' ',49X,'*** DEFINITION OF PARAMETERS ***' /I/I' ,28X,'BSUBIV - CONCENTRATION OF NUCLIDE PER UNIT ',
* 'FRESH WEIGHT IN PLANT (MICRO- I /* ' ',37X,'CURIES PER KILOGRAM) DIVIDED BY CONCENTRATION ',
*'OF NUCLIDE PER UNIT' /* ' ',37X,'DRY WEIGHT IN SOIL (MICROCURIES PER KILOGRAM)')
CISM 0019 WRITE (PTR,60)ISM 0020 60 FORMAT V ',28X,'FO - INITIAL GROUND DEPOSITION SOURCE ',
I (MICROCURIES PER SQUARE METER)'. /* ''92X,'FSUBF - THE FRACTION OF THE ANIMAL''S DAILY ',**INTAKE OF NUCLIDE WHICH APPEARS ' /* ',37X,IIN EACH KILOGRAM OF FLESH (DAYS PER KILOGRAM)./
* 'iT30,'FSUBM - FRACTION OF THE DAILY INTAKE OF ',
* 'NUCLIDE I BY A COW WHICH APPEARS' /* '',T39,'PER LITER OF MILK AT EQUILIBRIUM ',
* '(DAYS PER LITER).' I
CISN 0021 WRITE (PTR,70)ISM 0022 70 FORMAT (' ',28X,'LAMRR - RADIOACTIVE DECAY RATE OF THE NUCLIDE ',
* 'UNDER STUDY (PER DAY)'
CISN 0023ISN 0024
WRITE (PTR'80)80 FORMAT V(' 92?8X, 'TAUEXC - EXCRETION RATE OF A STABLE ISOTOPE OF '
58
°THF NUCLIDE FROM THE MUSCLE OF' /I- ' ',37X,tA STEER (PER DAY)'
C
ISN 0025 WRITE (PTP,90)ISN 0026 90 FORMAT (///I' ',SIX,'*** VALUES OF PARAMETERS *,*' //
* ' '~,28X, 'RSt)BTVV , ,1X ,IFO' , I X, 'FSUBF', 11X, 'FSUBM , IIX,
* 'LAMRR',IIX,'TAUEXC' /I
CISN 0027 00 1.10 = I,NUMNUCISN 0028 WRITE (PTR,100) NAMNUC(I),BSUBIV(1)tFO(I),FSUBF(I),FSUBM(f),
* LAMPR(I),TAUEXC( )
ISN 0029 100 FORMAT(17X,A8,6(El3.6,3X),/.)ISN 0030 110 CONTINUE
CC BRANCHING PATIOSCC
ISN 0031 IF (NUMNUC .EQ. 1) GO TO 35C
!SN 0033 WRITE (PTRt!OISN 0034 10 FORMAT ('1',54X,'BRANCHING RATIOS' I/I
* ' ',48X,'FRFjM',IX,'TO',9X,'FPACTION' //)
CISN 0035 00 30 I = 1,NUMNUC!SN 0036 DO 30 J = I,NUMNUCISN 00.37 IF (B(I,J) .FO. 0.01 GO TO 30ISN 0039 WRITr (PTP,20) NAMNUC(J),NAMNUC4I),IBI,J)ISN 0040 20 FORMAT (49X,A8,2X,A8,2X,E13.6,/lISN 0041 30 CONTINUEISN 0042 35 CONTINUE
CCC NUCLIDE INDEPENDENT PARAMETERSCC
ISN 0043 WRITE (PTR,1400 AISN 0044 140 FORMAT ('.l',5?X,'NUCLIDE INDEPENDENT PARAMETERS' //I
* ' ',27X,'A',6X,'- SOIL SURFACE AREA REQUIRED TO FURNISH ',* 'FOOD CROPS FOR, ONE ' /
',36X,'MAN (SQUARE METEQS)',38X,E13.61
CISN 0045 WRITE (PTR,1451 ASUBG
SIN 0046 145 FORMAT t ,27X, 'ASUBG - PASTURE AREA PER COW (SQUARE METERS)',* 21X,El3.6)
CISN 0047 WRITE (PTR,146) SMALLOISN 0048 146 FORMAT P' ',2TX,'SMALLD - DEPTH OF THE PLOW LAYER (CENTIMETERS)',
* 20X,E13.6 )
CISN 0049 WRITE (PTR,1471 RHOISN 0050 147 FORMAT (' ',2?X,'RHO - DENSITY OF THE SOIL (GRAMS PER CUBIC',
I CENTIMETERI',9X,E13.6C
ISN 0051 WRITE (PTR,150) DSUBGTSN 0052 150 FORMAT P' ',27X,'DSUBG - DRY WEIGHT AREAL GRASS DENSITY (KILO',
* 'GRAMS PER SQUARE'/S ' ' ,36X,'METER)',52X,E.13.6)
C
59
ISN 0053TSN 0054
ISN 0055ISN 0056
1 SI 0057IS5 0058
1SK 00591 SN 0060
(SN 0061I15 0062
I SN 006 3ISN 0064
1 SN 0065ISN 0066
ISN 0067ISN 0068
ISN 0069I SN •0070
1SN 0071ISN 0072
ISN 0073ISN 0074
ISSN 00751SN 0076
ISN 00771 SM 0078
ISN 0079ISN 0080
WRITE (OTR,165) MSURBIn5 FORMAT (' ',27X,'MSUJBB - MASS OF MUSCLE ON A STEER AT THE TIME
* 'OF SLAUGHTER' /* ' ',36X,'(KILOGRAMS PER STEER)',37X,E13.6)
C
WRITE (P'R,I90) TAUREF190 FORMAT (' ',27X, 'TAtUBEF - FRACTION OF THE BEEF HERD SLAUGHTERED
* ' (PER OAY)',!OX,EI3.6)C
WRITE (PTR,1951 TAUMLK195 F3RMAT (' ',27X,'TAUMLK - TRANSFER RATE OF MILK FROM THE UDDER
I
I ,
t '(PFR DAY)',11X,E13.61
WRITE (PTR,200) TAUES200 FORMAT (' ',27X,'TAUES - TRANSFER COEFFICIENT
'(PER DAY)',15X,E13.61
CWRITE (PTR,205) TAUGR
205 FORMAT (' ',27X,'TAUGR - TRANSFER COEFFICIENT'(PER DAY)',15X,E13 0 6!
C
WRITE (PTR,210) TAUPH210 FORMAT (' ',27X,'TAUPH - TRANSFER COEFFICIENT
A'(PER DAY)#,15X,Ei3.61C
WRITE (PTR,215) TAURD215 FORMAT I' ',27X,'TAURD - TRANSFER COEFFICIENT
* '(PER DAY)',15X,E13.6)C
WRITE (PTP,220) TAuRG220 FORMAT (' ',27X,'TAURG - TRANSFER COEFFICIENT
'(PER DAY)',15X,E13.61C
WRITE (PTR,225) TAUSP225 FORMAT (' ',27X,'TAUSP - TRANSFER COEFFICIENT
* '(PER DAY)',15X,E13.6)
CWRITE (PTR,235) U
235 FORMAT (' ',27X,'U',bX,'- MILK CAPACITY OF THE* 22X,E13.6)
C
FROM E TO S
FROM G TO R
FROM P TO H
FROM R TO 0
! $,
|9
FROM R TO G
FROM S TO 'n
UDDER (LITERS)',
WRITE (PTR,2401 VSUBC240 FORMAT P' ',27X,'VSUBC - DRY WEIGHT GRASS CONSUMPTION PEP ',
* 'DAY BY A COW (KILOGRAMS' /* ' ',36X,'PER DAYI',50X,E13.61
CWRITE (PTR,3001 TIMBH
300 FORMAT (' ',27X,'TIMBH - HOLDUP TIME (DAYS) FOR BEEF',31X,E13.6)C
WRITE (PTR,305) TIMCH305 FORMAT i' ',27X,'TIMCH - HOLDUP TIME (DAYS) FOR MILK',31X,E13.6)
CC OUTPUT SPECIFICATION TIMES
CC
WRITE (PTR,260)260 FORMAT 1'1',52X,IOUTPUT SPECIFICATION TIMES' //
* ' ',54X,'INCREMENT',4X,'END OF INTERVAL' /
60
* ' ',56X,'lDAY)',11X,'(DAYl' I/IC
ISN 0081 DO 280 1 = 1,NTNTVLISN 0082 WRITE (PTR,270) INCR(I),ENDTIM(I)!SN 0083 270 FORMAT (53X,E13.6,6X,161ISN 0084 280 CONTINUE
CC*** INFLOW RATES FOR VARIOUS SPECIES
ISN 0085 WRITE (PTR,400)[Sh 0086 400 FORMAT (1'1,20X,'INFLOW PATES FOR VARIOUS SPECIES')
CISN 0087 WRITE (PTR,4011ISN 0088 401 FORMAT ('O',26X,'INITIAL TIME',22XORATE')
CISN 0089 WRITE (PTR,402)[SN 0090 402 FORMAT (' ',8X,'NUCLIDE',14X,'(DAYS)',20X,•(UCI/SQ.M/DAY)II
CISN 0091 DO 406 1=1, NUMNUCISN 0092 WRITE (PTR,407)ISN 009.3 407 FORMAT(' 'IISN 0094 DO 406 KP=I, MPISN 0095 IF (KP .EQ. 1) WRITE (PTR,404) NAMNUC(I),TIMEP(KPI,FF(I,KPlISN 0097 IF (KP .GT. i1 WRITE (PTR,405) TIMEP(KP),FFUI,KP)ISN 0099 406 CONTINUEISN OLO0 404 FORMAT (' ',8X,A8,9X,E10.3,18X,EIO.31[SN 010.1 405 FORMAT (I ',25X,EIO.3,18X,EIO.31
CC GEAR SUBPOUTINE PARAMETERSCC
ISN 0102 WRITE (PTR,41O|ISN 0103 410 FORMAT ('1',54X,'GEAR SUBROUTINE PARAMETERS')
C
ISN 0104 WRITE (PTR,420) EPSISN 0105 420 FORMAT ('O',55X,'EPS = ',013.61
CISN 0106 WRITE (PTR,421) MFISN 0107 421 FORMAT (1O',T57,'MF = ,12
CISN 0108 RETURNISN 0109 END
61
LEVEL 21.8 f JUN 74 1 OS/360 FORTRAN H
ISN 0002
1S14 0003ISN 000OISN O0O05
ISN 0006
ISN 0007ISN 00O8ISN 0009
ISN 0010ISN 0011ISN 0012
COMPILER OPTIONS - NAME= MAIN,OPT=02,LINECNT=60,SIZE=OOOOK,SOURCE ,EBCDIC, NOLI.STNODECK, LOAD, MAP, NOEDIT, ID, NOXREF
SUBROUTINE CALCIN
CCCC CALCULATES THE VALUES OF CERTAIN PARAMETERS WHICH ARE USED IN
C DEFINING THE COEFFICIENTS OF THE SYSTEM OF DIFFERENTIAL EQUATIONS.
CCC
COMMON /TIME/ TIM(365),INCR(3O),ENDTIM(30),NINTVL,tNTIMCOMMON /CROPS/ MFOCOMMON/LMDA/LAMR415),LAMIC(15),LAMB(15),LAMA(15),
C LAMS( 15),LAMG( 151COMMON /INDEP/ A,ASUBG,,SUn..,MSUBB,TABEF,TAUMLKTTA'UESTA-UGR,
* TAUPH,TAIJRD,TAIJPRGTAUSP,UVSUBC ,SMALLD,RHO
COMMON /NUMBRS/ NUMNUCCOMMON /PARAM/ TAUGC(15),TAUGB(15)
COMMON /DEP/ LAMRR(15),FSUBM(15),TAUEXC(15),FSUBF(15),BSU.BIV(15)CCC
DOUBLE PRECISION TIMINTEGER ENDTIM
REAL MSUBB,LAMRP,INCR,LAMR,LAMC,LAMB,LAMA,LAMS,LAMG,MFOC
C MFO = TOTAL MASS (GRAMS) OF CROP AT HARVEST TIME - USED IN SUBROUTINES
C CALC AND TIMDEP (BASED -N A SQUARE METERS OF LAND, 250 PLANTS PER
C SQUARE METER, i GRAM PER PLANT).C
MFO = A * 250.0CC
00 1 T=I,NUMNUCTAUGC(I)= (TAUMLK*FSUBR(MI*VSUB') / DSUBGTAUGB(I)= (VSUBC*FSUBF(I)*TAUEXC(I1) / DSUBG
ISN 001o
ISN 0014ISN 0015ISN 0016
CC
ISNI SNI SNISNI SNISNISN
0017001800190020002100220023
DEFINE LAMDASLAMR(II= LAMRP(I)LAMC(I)= LAMRR(I)LAMB(I)= LAMRR(I)LAMA((J= LAMRR(IILAMS(I)= LAMRR(I)LAMG(I)= LAMRR(I)
I CONTINUE
÷4.÷
÷
4.+
TAURG + TAURDTAUMLKTAUBEF + TAU(TAUESTAUSPTAUGR + VSUBC,
EXC(I)
/(ASUBG*DSUBG)
CCCCCCCCCC
DEFINE TIME ARRAY FROM THE FOLLOWING INPUT INFORMATION.
NINTVL = NUMBER OF SUBINTERVALS.
ENDTIM(I) = THE RIGHT ENDPOINT OF SUBINTERVAL I.INCRMI = THE TIME INCREMENT FOR SUBINTERVAL I.NPREV = 0NTIM = 0
ISN 0024ISN 0025
62
C
ISN 0026 DO 2 I=I,NINTVL!SN 0027 \11 = 1-1ISN 0028 IF(I.EQ.Il STAPT=0.0ISN 0030 IF(I.NE.1) START= ENDTIMIMI)ISN 0032 NUMSUB = (ENOTIM(I)-STARTI / INCR(1)ISM 0033 NTIM = NTIM + NJUMSUB
cISN 0034 DO 3 J=1,NUMSUP,ISN 0035 II = NPPEV + JISN 0036 3 TIM( If) = START + J*INCR(1)
ISN 0037 NPREV = NUMSUil + NPREVISN 0038 2 CONTINUE
CC
ISN 0039 RETURNi SN 0040 SNO
63
LEVEL 21.3 JiUN 74 ) 0S/360 FORTRAN H
COMPILER OPTIONS -NAME= MA!N,OPT=O2,LINECNT=60,SIZE=OOOK,SOURCF, FBCDI C, tOL I ST, NODECK,LOAD, MAP,NOEDIT, I D, NOXREF
ISM 0002 SUBROUTINE CALCCCC
C CALLS GEAR SU9ROUTINE TO CALCULATE AN' PRINT VALUES OF STATE VARIABLESCr.
"' C
TSN 0003 EXTERNAL PEDEPV,DIFFUN
ISM 0004 DIMENSION TR(15),TBf(15,PRORAT(15),OOBEF(15),QOMLK{(15),* QBEF(15),QMLKL15) ,QWIGL(I5)
'IN 0005 COMMON /OEP/ LAr4RR(-15),FSUBM(15), TAUEXC( 15),FSUBF( 15) ,SUBIV(15)ISN 0006 COMMON /!,RANCH/ B(15,151iSN 0007 C7]MMON,4 /HOLTIM/ TIMRH,TI-l.C,ISN 3008 COMMON /GPAPAM/ EPS,MFTSN 0009 COMMON /MCHECK/ TOTUCI(365,15)ISN 0010 COMMON /NUMBRS/ NUINUCISN 0011 COMMON /TIME/ TIM(365),INCR(30),ENDTIM(30),N!NTVLNTIMISN 0012 COMMON /SOURCE/ FO(151!SN 0013 COMMON /NAMES/ NAMNUC151ISN 0014 COMMON /INDEP/ A, ASUBG, DSUBG,MSUBB,TAUBEF,TAUMLK, TAUES, TAUGR,
* TAUPH,TAUROTAURGTAUSP,U,VSUBC,SMALLD,RHO
ISM 0015 COMMON /HRVST/ HAPTIM(30),NHARVTSN 0016 COMMON /CROPS/ MFO
ISN 0017 COMMON /TOEV / OTR,RDR
CC THE DIMENSION OF THE ARRAY YO MUST BE AT LEAST 12*NUMNUCC
ISM 0018 DOUBLE PRECISION LOG2,TIM,TOUT,YO(72),NAMNUC,TO,HO,EPSTIME,* HARTIM
ISN 0019 INTEGER ENDTTM,PTR,RDRISN 0020 REAL LAMRR,INCR,MFO,MSUBB
C
ISN 0021 DATA [DIM 115/CC
ISM 0022 WRITE (PTR,I00)ISN 0023 100 FORMAT ('1',T55,'RAGTIME COMPARTMENTS' //
CISN-0024 WRITE (PTR,105)ISM 0025 105 FORMAT (' ',28X,'E - RADIOACTIVITY PRESENT ON ABOVE-SURFACE ',
'FOOD PER SQUARE METER OF SURFACE' I* ' ',34X,'ON WHICH FOOD CROP IS GROWN (MICROCURIES PER ',
* 'SQUARE METER).')
ISM 0026 WRITE (PTR,107)ISN 0027 107 FORMAT ('0',28X,'S - RADIOACTIVITY PRESENT AT THE SOIL ',
'SURFACE (MICROCURIES PER SQUARE METERS.')C
ISN 0028 WRITE (PTR,110)ISN 0029 110 FORMAT ('O',28X,'P - RADIOACTIVITY PRESENT IN THE SUBSURFACE '.
'POOL ASSOCIATED WITH ONE MAN''S'/* ' ',34X,'FOOD SUPPLY (MICROCURIES1.'I
CI SN 0030ISN 003 1
WRITE (-PTR-lR5R115 FORMAT 113-',28X,'G - RADIOACTIVITY PRESENT IN THE GRASS CON',
64
• 'PPAPTMENT (MICROCURIES PER SQUARE' IS',34X,' METFR).')
CISN 0032 WRITE (PTR,1201ISN 0033 120 FORMAT 0'0',28KR - RADIOACTIVITY PRESENT IN THE SOIL FROM ',
"* 'GROUND SURFACE TO THE ROOT DEPTH' /c ' ',34X,'flF THE GRASS (MICROCURIES PER SQUARE METERI.')
C
ISM 0034 WRITE (PTR,I30)ISN 0035 .130 FORMAT {'0',28X,'C - CONCENTRATION OF RADIOACTIVITY IN THE ',
*• 'MILK (MICROCURIES PER LITER).')
CISN 0036 WRITE (PTR,140)ISN 0037 140 FORMAT ('0',28X,'B - CONCENTRATION OF RADIOACTIVITY IN THE ',
C Ix 'BEEF (MICROCURIES PER KILOGRAM).')C
TSN 0038 WRITE (PTR,150).S .. 0039 1 .0 FORMAT (102,28XT - RADIOACTIVITY PRESENT IN THE INTERIOR ',
*'OF PLANTS PRODUCED FOR HUMAN CON-#/* ' ',34X,'SUMPTION (MICROCURIES).')
CISN 0040 WRITE (PTR,152)ISN 0041 152 FORMAT ('O',Z8X,'EH - CONCENTRATION OF RADIOACTIVITY IN FOOD ',
* 'WHICH IS STORED FOLLOWING' /* ' ',34X,'HARVEST OF CROPS (MICROCURIES PER KILOGRAM).')
ISN 0042 WRITE (PTR,154)ISN 0043 154 FORMAT f'O',28X,'BH - CONCENTRATION OF RADIOACTIVITY IN BEEF '
'HOLDUP COMPARTMENT (MICROCURIES' /' ',34X,'PER KILOGRAM).')
CISN 0044 WRITE (PTQ,156)ISN 0045 156 FORMAT ('0',28X,'CH - CONCENTRATION OF RADIOACTIVITY IN MILK ',
'HOLDUP COMPARTMENT tMICROCURIES' /*= ' ',34X,'PER LITER1.')
CCCC
C INITIALIZE STATE VARIABLESC Y(I)=E, Y(2)=S, Y(3)=P, Y(4)=G, Y(5)=R, Y(61=D, Y(7)=C, Y(81=B,
C Y(9)=EH, YI!O)=T, YI|l)=H, Y(12)=MPRIMECCCC DETERMINE S VALUES AT TIME ZERO.
ISN 0046 TZERO = 0.0ISN 0047 CALL SVAL(TZERO,SlS2,S3,S4)ISN 0048 DO 5 I=1,NUMNUC
C NPREV= NO. OF PREVIOUS STATE VARIABLES. FO(I)=DEPOSITION SOURCE (NUCLIDE I).ISN 0049 NPREV = (1-1i*12ISN 0050 YO(NPREV+1)=S1*FO(I)ISN 0051 YO(NPREV+2)=S2*FO(I)ISN 0052 YO(NPREV÷3)=O.ODOISN 0053 YO(NPREV+41=S3*WFO(I1ISN 0054 YO(NPREV+5)=S4*FO(I)ISN 0055 YO(NPREV+6)=0.ODOISN 0056 YO(NPREV÷7)=.0DO
65
ISN 0057 YOINPREV+8)=O.ODOISN 0058 YO(NPREV÷9)=O.ODOISN 0059 YO(NPREV+1O)=O.ODOISN 0060 YO(NPREV+11J=O.OOISN 0061 YO(NPREV*12)=O.ODOISN 0062 5 CONTINUE
CC INITIALIZE BEEF AND MILK HOLDUP COMPARTMENTS BH AND CH RESPECTIVELYCC QBEF{II = BURDEN OF NUCLIDE INUC IN COMPARTMENT BH (UCI / KGI.C QMLK(I) = BURDEN OF NUCLIDE INUC IN COMPARTMENT CH (UCI / L ).C
ISN 0063 DO 350 I=I,NUMNUCISN 0064 QBEF(I) = 0.0ISN 0065 QMLKIII = 0.0ISN 0066 350 CONTINUE
CCC DEFINE GEAR SUBROUTINE PARAMETERSCC
ISN 0067 INDEX = 1C TO= INITIAL TIME (DAYI.
ISN 0068 TO = O.ODOC HO= NEXT STEPSIZE IN T (DAYS), USED ONLY ON FIRST CALL.
ISN 0069 HO = 1.0D-6CCC CALCULATE AND PRINT VALUES OF ES,P,GR,C,B,TEH,BHCH AT TIMESC TIM(II, I=INTIM.CCC N=NUMBER OF EQUATIONS.
TSN 0070 N= NUMNUC*12CCC *** PAGING DESCRIPTION ***
CC KOUNT IS THE NUMBER OF OUTPUT SPECIFICATIONS TO BE PRINTED PER PAGE.C ITS VALUE IS DETERMINED BY USING AN INTEGER FUNCTION OF THEC NUMBER OF NUCLIDES IN THE CHAIN BEING STUDIED PLUS ONE FOR SPACINGC DIVIDED INTO THE TOTAL NUMBER OF AVAILABLE LINES PER PAGE AFTERC HEADINGS HAVE BEEN PRINTED. K IS INITIALLY SET EQUAL TO KOUNT9C THUS ENABLING THE HEADINGS TO BE PRINTED THE FIRST TIME THROUGHC THE LOOP. ONCE THE HEADINGS ARE PRINTED K IS SET TO ZERO AND ISC INCREMENTED BY ONE ON EACH PASS THROUGH THE LOOP UNTIL K IS EQUALC TO KOUNT. THEN THE PROCESS OF PRINTING THE HEADINGS AND SETTINGC K EQUAL TO ZERO CONTINUES.CC
[SN 0071 KOUNT = 58 / (NUMNUC + 11ISN 0072 K = KOUNT
CCC DEFINE PARAMETERS FOR CALL TO RESDNS. THE FOLLOWING PARAMETERSC ARE NOT TIME-DEPENDENT AND ARE THEREFORE DEFINED BEFORE ENTERING THEC [TIM LOOP.C
66
C-
E-
T•q 1, <7 N>
TV1 007T 8I[ "'N " 0079
im 008
"15N 9085O
15N 0096•f5N 00 944
"rSN O0395
<SN 0)96
lSN 0398
1SN 0:199
1 SN 01,90FSN 10(I9
1SA 01046
1S', 310114•:q,9 1,
31N 0108
SN .0109
7P 1 00 1 =Pd D..8- ::F:F (DAY I OF =rCL " ('N (0C -1NNNNI F N -UFJ -' 10 1$)F I FF (nA OF INUC IE ¶1-UC. USE LAI> GG4 VALUE7 17
APP• Cq AT A BID, > F*OV At FAC TOR 1F '(749
L "..Z••D I NUC - ,N 0 M[O"100 1 So . IO, NC.MCNN C
, A" 4 jAU[]'.iC 4, N ,55fT45>T ?>14<)>> 'A> CN PA (7 F9 NICE~~ 1775 11(7
0 41[• A 'NC lJ> !=0
D,0 4 '174 (> It<IF IK NE'i. KOUNi T1!l< G"O" 7" 16,>)
r DOINT•' H ADJItNC
WRITE ¶)3t>P,Ij CNAMNt'ICIJI . NNNJNAU4t
i >R.,AT I11CN)NTENTS OF C0 NPARTM>T.S AT VA $1U US TISE.'S i,. NUCI.DES IN THE CHAIN... , 14&9)
I FORMNAT( 910 1 ' !M T 17, ' 51 .7 T ' I SII' 9v C' P 3 t N *S 0 t. ' G 6 ,T7 2, C
W9•!TE QpT0,313 (7 R-4.[I 1C.),4YSJ ,IO, (I•tCCi/<2r',1 T-
24 ftU /50.1) , 137,' iCC Pt
4C., 79 UL11 SC1I1,. .T.3,T " UCII. lS 4 4Tj 4, 1t/UCjjK l' .,TP30,'(U/ULS KI'(1... )T T(NNil "r i) , tClfib;l 3 T ,711¾' (10 /KG) ,T ,;zS% '<I~l, ....... /13
c IF T 1 3 PRINT VA7 ..LUEj S W!t> NMfl ll7:N> T ' AT T!M2,F1 Z7 9.,
2 : r tI 1 ($(H4[)423t>T !C~f'1i4 119 .[4>111 494<4{ tN4>j.3lYO(44,4fl4$4,Y*)ffl,
90(5 4 414 19$ CHUN''44I31(101 201 C8 .101 17O 'S 23' ]N FI N$> NN> (Cl ( PC.>' 1> '*4 SIN •0
NPRE N 12 1 T'N.C-. 1.)WRIT: ( T1..1 1 YOINP -NV4 )I,YOI4NP CfhVl'N'YONNI P 'EV *3),
> C: P V*4, GOI"NP $6V45 •Yt (N PP " ), V YON -PPV9* ),* Y( 0 t>P.V+1J. O 0 (NlfPRF{V&9e P , /4)!F $431 >(7NUC I ••L(IIJ
+ h" CONTINUE I - K t ACtJT f•R •
~ 1~<2 901 7 5 0 409197 , 4323!N INC !NO N.]PAfCTM 0NT VAI.LUES
o <T TI9N 5 0R,
TOUT = TfiA(1
CALL tGEAR 0FFUNPnERVN., SO H0,YO.TOQ TE7 ,M1,S•.,-TNEX),IF NICOEXE:.0) GSO TO 6R ITE (PT',11 TOW*'5 IND1E,
I' TORA • >1i47•C. WAS NOT CfOlMPLE TED >TO 1OWT9 't1,L1O.3,.' , INDE1X= 1.
GO to. 9
]
5)
cr.
cC,c
C.C
TNE F9LLOW1WNG STATTMENTS ARE USE(7D TO LCILATE THE TOTAL AM4UNT 1 FCI 40C1TV0TY I9MIC(0C9U0IESI IN iHE SYSTEM FOR EACH NUCLIE 14 T74r
TOTUCtII,JNUtCI = RADIOACTIvTTY FOR$ NU.tCLI E W4ITH INDEX JNUC
67
PrMQ~ OV.0 SCA OZOULEIo UIS!(OW £TEMA-UIATIN9 C WECK.4
iS%1•S sI
¶44~
01120112OU12OiII0!.[4
ý,9N'IT ¶102INDEX - 2
D0 20 xIU- .) ,IPNUINlPREY = 12 * {JmI c-ll
PI<!T UCIW,'I' . )(N21
1v41)OA * V PRF0VI0 2)*A 4 Yo11P60V43}
YOINU P 6 L )*ttIUttO I I~ K$ WVIIASURC + Y0IW7D'V!- WWSW',G
20 K.•YTtnIC ":
C5 ~ ~ ~ ~ ~ ~ y( -A< K '0'0Y0;C A ME 5QpCAL p AMPQ7
C
C 1, IN0=0).!"14A1 30EN Q NUCI..IE')fN IUIJC IN BEEr CYIMPARTMKNT.
C fQY0flIK(!%UJC Im=U ITfAL -041 F N1L)CLPF I-'1. 0 C IN MILK COnMPRTME-NT.
09 301 !NUC=U .UM -IC
C F / I TINCI Y n 2,
W NU4. Y F1 ý) f
lit mE fl 1tI,1INAVI, R,T•3,S,)SPTOTLKOL¶14'C1DM)
,I S"
IT SI
0116
3117011.1?
0 (1t31/31121
112 2
I,C
0>4INT VALU}E aR4TME N S AT TIM•(!I )
;N 0123 WRITE (PTR, .i) TOUT,Y.Ivt) ,YO(1LYOO31,Y0(4),Y015),YO7),
VIF (I I FY0 ) C0| 70(91 ,G ,F T I t MLK70 1
1 F. HIUINII2
, <>. 4• 00; TI 1,70
'f" VALUES Qf v .0,Cs,fT. I, E ,OH CH FOPR -EM•tIl1 i; HUOL !ES.
oCt 01271sm 0123
Sm. 9130(SN•' iIt
SN 01.32
I 3N II} 33•I SIK 01 t34
f 5 0156iSN 0137
(SN 0139. S j OL39ISN 0140
Np2RF V= 12 11.K N UI-IN0
WRITE IPTR121 YU(NPAIKV+I),YO(N.'PREV42),,YO{NPREVf3),c Y.0(NPREV+t4,YOI{NPKIýgN* ¶ID+5 Y0NPN. EV+/}), Y0ONPREVt8}),
Y y ( N PREV*IO 1 ,YO I1NPRCV*+9 ) , SB EF ( IN1UC ) ,QM LK ( INUC I4 CONTINUE
12 FORMCAT" 4 • .I, 11 I010.3.' '"I
120 CONTINUEC IS 'IMI I A H4ARVEST !I',? QUERY PETOANS I IF YES, 03 IF NO.
CALL QUERYI(TI4 IANI )IF (4ANS.00.01 0. T0 171
C AT HARVEST TINE, REINTAL.tIZf STATE VARIASLES.TO TI III)HO 1.00-5INDEX = I00 250 1NUC=I , ,UMNICNPREV a f(IlNC-1V*12
CC FOR THE PURPOSE OF MASS BAt ALCEN CHECK, TRANSFER TOTAL ACTIVITYC FR.M CROP HCLDUP CPMPARTMENT (EHO TO COMPARTMENT MPPIME,
68
TSN 0141 YO(NPREVd12) = YO(NPREV+12) + (MFO * .001) * YO(NPREV*9)CC DIVIDE TOTAL ACTIVITY (UCI) TRANSFERRED TO CROP HOLDUP COMPARTMENTC EH FROM E AND T BY TOTAL MASS (KG) OF CROP AT HARVEST.
ISN 0142 YO(NPREV*9) = (A*YO(NPREV+I) + YOiNPREV+1O)) / (MFO * .001)ISM 0.143 YO(NPREV+l) = O.ODOISN 0144 YO(NPREV+I,O)=.O.ODOISN 0145 250 CONTINUEISN 0146 171 K=K+÷ISN 0147 4 CONTINUE
CCCC UPON COMPLETION OF I LOOP, RETURN.
ISN 0148 GO TO 10C IF GEAR SUBROUTINE RETURNS INDEX OTHER THAN 0, STOP.
ISN 0149 8 STOPCC
ISN Olo 10 RETURNISN 0151 END
69
LEVEL 21.8 ( JUN 74 1 0S/360 FORTRAN H
COMPILER OPTIONS - NAME= MAINOPT=02,LINECNT=60,SIZE=OOOOK,SOURCE,EBCDIC,NOLISTNODECK,LOADMAP,NOEDIT, ID,NOXREF
ISN 0002 FUNCTION F(I,TjC F4I,Tl = SOURCE STRENGTH (UCI/SQ.M/DAY) FOR NUCLIDE I AT TIME T (DAYS).CC F IS DEFINED IN TERMS OF THE INFLOW RATE MATRIX,C
C FF(I,KPI , I=1,NUMNUC, KP=1,MPC
C WHICH IS DEFINED IN SUBROUTINE INPUT.ISN 0003 COMMON /INFLOW/ MP, TIMEP(30),FFHI5,301ISN 0004 COMMON /NAMES/ NAMNUC(151ISN 0005 DOUBLE PRECISION NAMNUCISN 0006 F = 0.0ISN 0007 IF (T.GE. TIMEP(MP) .OR. MP.EQ.l) GO TO 3ISk 0009 MPMI = fvP-1ISN 0010 DO 1 KP=I,MPMIISNOOlI IF (T.GE.TIMEP(KP) .AND. T .LT. TIMEP(KP+1) GO TO 5ISN 0013 1 CONTINUEISN 0014 WRITE(3,2) NAMNUCtI), TISN 0015 2 FORMATf'OF(I,T) NOT DEFINED FOR NUCLIDE ', A8,' TIME ',EIO.3,
C ' (DAYS)' )ISN 0016 STOPISN 0017 5 F= FF(I,KP)ISN 0018 GO TO 4ISN 0019 3 F=FF(I,MP)ISN 0020 4 RETURNISN 0021 END
70
LEVEL 21.8 ( JUN 74 O OS/363 FORTRAN H
COMPILER OPTIONS - NAME= MAINOPT=O2,LINECNT=60OSIZE=OOOOK,SOUPCE,EBCOIC,NOLISTNO)ECK,LOOD,M.P, NOEDIT, IO,NOXREF
ISM 0002 SUBROUTINE O)IFFUNIN,T,Y,YDOT)CCCC COMPUTES THE RIGHT HAND SIDE OF YDOT=F(Y,T)CCC
ISM 0003 O[MENSII]N TAUPT{15)ISN 0004 COMMON /INOEP/ A,ASUEG, OSUBG,MSUJBB, TUBEF,TAUMLK,TAUES,TAUGP,
*TAUPH,TAiJRn,TAURGTAUJSPU,VSUBCSMALLD,PHO
ISN 0005 CONIMON/LMDA/LARI(15),LAMC(15hLAMB(15S,LAMA({15),C LAMS(15 ,LAMG( 151
ISN 0006 COMM ON /iEP/ LAMRR( 15),F-SUBMIi5),TAUEXC( 15),FSUBFf 15),BSUBIV 15 )ISN 0007 COMMON /PARAM/ TAUGC(15),TAUGBfIl5ISM 0008 COMMON !NUMBRS/NUMNUCISN 0009 COMMON /BRANCH/ 8(15,15)ISN 0010 COMMON /1OEV / PTR,ROR
CC
CISN 0011 DOUBLE PRECISION T,Y(NI,YDOT(N)ISN 0012 REAL L AMBH, LAMCHMSUBB, LAMRR, I ILAMR,LAMC, LAMB, LAMA, LAMS,
* LAMG,IIPRIM,LAMPIISN 0013 INTEGER R0RPTR
CCCC THE STATE VARIABLES AREC Y(NPREV+I-) = EC Y(NPREV+2) = SC Y(NPREV,31 = PC Y(NPREV-4) = GC Y(NPREV+5) = PC Y(NPREV+bI = 0C Y(NPREV+7) = CC Y(NPREV+81 = BC Y(NPREV+91 = EHC YINPREVGOIO = TC YINPREVt11) = HC Y(NPREV+12) = MPRIMEC WHERE NPREV = 0,,2,...,NUMNUC-1CCCC TIME (T) IS SOMETIMES REQUIRED TO BE IN SINGLE PRECISION (XM.
ISN 0014 X=TISN 0015 CALL SVAL(X,SI,S2,S3,S4)ISN 0016 CALL TIMDEP(X,TAUPT)
CISN 0017 DO I I=1,NUMNUC
C NPRFV = NO. OF PREVIOUS EQUATIONSISN 0018 NPREV= (I-11*12
C CALCULATE CCNTRIBUTION FROM PREDECESSORS IN THE CHAIN.ISN 0019 SUMI = 0.0ISM 0020 SUM2 = 0.0
71
ISNI SNISNI SN
SI j
I SNI SNI Sm
I SNI SNI SNI SN
002100220023002400250026002700280029003000310033
ISN 0034ISN 0035!SN 0036TSN 0037ISN 0038ISN 0039ISN 0040ISN 0041T!SN 0042
I SN 0043I SN C044TSN 0045ISN 0046ISN 0047
ISN 0048
I SN 0049ISN 0050
ISN 0051!SN 0052
ISN 0053ISN 0054
ISN 0055
ISN 0056
ISN 005.7
ISN 0058
ISN 0059
SUM3 = 0.0SUM4 = 0.0SUM5 = 0.0SUM6 = 0.0SUM7 = 0.0SUMS8 = 0.0SUM9 = 0.0SUmlO = 0.0SUM I1 = 0.0SfJM12 = 0.0IF (I.LE.1I GO TO 3IMI= 1-I
DO, 2 J=l, IMIJPREV=1 J-1)*12
SUM, = SUMI + LAMRR(I)*B(I,J)*Y(JPPEV+I)SUM2 = SUM2 + LAMRR(I)*BlI,JJ*Y(JPREV÷2)SUM3 = SUM3 + LAMRR(I)*B(I,J)*Y(JPREV÷3)SUM4 = SUM4 + LAMQR(I)*B(IJ)*Y(JPREV+4)SUM5 = SUM5 + LAMRR(II*B(IJ)*YIJPREV÷51SUM6 = SUM6 + LAMRR(I)*B(IJ)*Y(JPREV+6)SUM7 = SUM7 + LAMR.R(1I*B(IJI*Y(JPREV+7÷SUM8 = SUM8 + LAMRR(I)*B(IJI*Y(JPREV+8)SUM9 = SUM9 + LAMRR(I)*BIIJ)*Y(JPREV+9)SUMIO = SUMlO +LAMPR(I)*B(I,J)*Y(JPREV+10ISUM1I = SUMil +LAMRR(1I*8(I,J)*Y(JPREVtll)SUM12 = SUM12 +LAMRR(Il*B(I,JI*Y(JPREV÷12)
2 CONTINUECC CALCULATE YDOT
3 YDOT(NPREV+1)=SI*F(I,XJ-LAMA(Ij*Y(NPREV+I) + SUM1YDOT(NPREV+2)=TAUES*Y(NPREV+1)-LAMS(I )*Y(NPREV+21+S?*F(I,X)÷SUM2
C LAMP! IS TIME DEPENDENT.LAMP, = LAMRR(I)+TAUPT(II + TAUPHYDflT(NPREV+3)=A*TAUSP*Y(NPREV÷2)-LAMPI *Y(NPREV÷3)+SUM3YDOT(NPREV+4)=S3*F{I,XI-LAMG(Ii*Y(NPREV+4)+TAURG*Y(NPREV+S)+SUM4YDOT(NPREV+51= S4*FtI,Xl +
C TAUGR*Y(NPREV+41-LAMR(II*Y(NPREV+5[*SUM5YDOT (NPREV+6) =TAURD*Y( NPREV+51
C -LAMRRII)*YINPREV÷6)+SUM6YOOT(NPREV+T)=TAUGC( II
C *Y(NPREV+4)-LAMC( I 3*Y {NPREV÷71÷SUM7YDOT(NPREV+8)=TAUG( I)
c *Y(NPREV+4)-LAMB(II*Y(NPR EV+8) +SUM8C
YDOT(NPREV+9) = -LAMRR(II * YINPREV+91 + SUM9C
YDOTINPREV÷iO)=TAUPT(I| * Y(NPREV+3) - LAMRR(I) * Y(NPREV+10)C +SUMIO
CYDOT(NPPEV4-11) = tTAUPH/A)*Y(NPREV+3) -LAMRR(I)*Y(NPREV*Ill
C THE COMPARTMENT MPRIME (Y(NPREV+121) RECEIVES ALL RADIOACTIVITYC FROM 8 AND C. THE TRANSFER COEFFICIENTS FROM B AND C TO MPRIMEC ARE TOTBM AND TOTCM RESPECTIVELY AND ARE DEFINED AS FOLLOWS.
ISN 0060
72
ISN 0061 TOTBM=TAUSEF*MSUBBISN 0062 TOTCM=TAUMLK*UISM'J 0063 IIPRIM = TOTBM * Y(NPREV*8) + TOTCM * Y(NPREV+7)ISN 0064 YDOT(NPREV+121 IIPRIM -LAMRR(I)*Y(NPREV+1,2) +
C (VSUBC/DSUBG - TAUGB(I! )*MSUBB - TAUGC.(I)*U) * Y(NPREV+4)
C + SUMi12C
TSN 0065 1 CONTINUECC
!SN 0066 RETURNISN 0067 END
73
LEVEL 21.8 t JUN 74 1 OS/360 FORTRAN H
COMPILER OPTIONS - NAME= MAIN,OPT=O2,LINECNT=60,SIZE=OOOK,SOURCE,EBCDIC, NOLIST ,NODECK,LOAD,MAPNOEDITID,NOXREF
ISN 0002 SUBROUTINE PEDERV(N,T,Y,PfNO)CC
ISN 0003 DOUBLE PRECISION T, Y(NO,131, PD)CC
ISN 0004 RETURNISN 0005 END
74
LEVEL 21.8 ( JUN 74 ) 0S/360 FORTRAN H
COMPILER OPTIONS - NAME= MAINOPT=O2,LINECNT=60,SIZE=OOOOK,SOURCE,EBCDIC,NOLIST,NODECK,LO D,MAP,NDED IT,ID,NoXREF
ISN 0002 SUBPOUTINE CHECKC CALCULATES AND PRINTS VALUES OFCC TOTUCI(ITIM,!NUC) = TOTAL ACTIVITY DUE TO NUCLIDE INUC AT TIME ITIMC AS CALCULATED BY SUBROUTINE CALC USING THEC GEAR SUBROUTINEC ACT(INUr,ITIM) = TOTAL ACTIVITY DUE TO NUCLIDE INUC AT TIME ITIMC AS CALCULATED BY SUBROUTINE TRAFUN, USING THEC BATEMAN EQUATIONSC AS WELL AS THE PERCENTAGE EPRnRCC (TOTUCI(ITIM,INUC) - ACT(TNUC,ITIMIU*100. / ACT(IN'ICITIMICC FOR TIMES TIMCITIM), ITIM=1 TO NTIM.r
ISN 0003 DIMENSION TR(15),TB(15),ACT(15,365),P(15,30I,RELEP.(151,RTIM(3651,*AWIGL(15,365)
ISN 0004 COMMON /INFLOW/ MP,TIMEP(30),FF(15,301ISN 0005 COMMON /BRANCH/ B(15,15)ISN 0006 COMMON /MCHECKI TrTUCI(365,151ISN 0007 COMMON /DEP/ LAMPR(15),FSUBM(151,TAIJEXC(15),FSUHF(15 ,BSUI1V(15)iSM 0008 COMMON /INDEP/ AASUBG,DSUBGMSUBB,TAUBEFTAUMLK,TAUESTAUGR,
* TAUPHTA'JRDTAURG,TAUSP,U,VSUBC,SMALLD,RHOI SN 0009 COMMON /NUMBRS/ NUMNUCISN 0010 COMMON /TIME/ TIM(365?,1NCR(30),ENDTIM(3O),NINTVL,NTIMISN 0011 COMMON /IODEV / PTR,RDR
CISN 0012 DOUBLE PRECISION TTM,LOG2-SN 001-3 INTEGER ENDTIM,PTR,RDRISN 0014 PEAL MSUBB,INCR,LAMRR
C IDIM IS THE MAXIMUM FIRST D!MENSION FOR THE ARFAYS ACT, P, AND AWIGL.C IT CORRESPONDS TO THE MAXIMUM NUMBER OF NUCLIDES IN A CHAIN.
ISN 0015 DATA IDIM /15/CC PRINT HEADINGS FOR TIME, TOTAL ACTIVITY COMPUTED BY THE GEARC SUBROUTINE, TOTAL ACTIVITY COMPUTED USING BATEMAN EQUATIONS, ANDC RELATIVE ERROR.C
ISN 0016 WRITE (PTP,100)ISN 0017 100 FORMAT ('I',52X,'COMPARISON OF TOTAL ACTIVITY' /
* '0',1IOX,'TIME (DAYSI',9ZX,'TOTAL ACTIVITY (MICROCURIESI',* 12X,'TOTAL ACTIVITY (MICROCURIFS)',8X,'PERCENTAGE ',
* 'ERROR' /*I ',35X,'*** GEAR SUBROUTINE ***',17X,'*** BATEMAN ',
* 'EQUATIONS ***'?CCC DEFINE INPUT PARAMETERS FOR SUBROUTINE TRAFUNCC TP(INUC) =RADIOACTIVE HALF-LIFE (DAY) OF NUCLIDE I.C TB(INUC) =BIOLOGICAL HALF-LIFE (DAY) OF NUCLIDE I (USE LARGE VALUEC TC APPROXIMATE A BIOLOGICAL REMOVAL FACTOR OF ZERO.)C P(INUC,KPI=SOURCE STRENGTH (MICROCURIES/DAYI FOR NUCLIDE IC FROM TIMEP(KPI TO TIMEPLKP+I) IF KP .LT. MP AND P(INUCMP=C SOURCE STRENGTH AT TIMES SUBSEQUENT TO TIMEPIMP).C BRANCH(I,J)
75
C =RRANCHING PATIO FROM SPECIES J TO SPECIES I, J LESS THAN I.C =B(I,Jl (INPUT DATA - C3MMflN/3PANCH/
CI SN 0018 L0,G2= DLOG(2.0DO)ISN O0!9 DO I INUC=I,NUMNUCISN 0020 TR(INUCI = LOG2 / OBLE(LAMRR(INUCJIISN 0021 TB(INUC) = 1.OE50
!SN 0022 DO0 6 KP=IMPYSN 0023 CALL SVALLTIMEPfKP),S1,S2,S3,S41
ISN 0024 6 PIINUCKPI = FF(INUC,KP) * ((SI÷S2)*A + (S3+S4)*ASURG)
!SN 0025 1 CONTINUECC RTIM IS A REAL ARPAY WHOSF VALUES ARE THE SAME AS THOSE FOP TIM.
ISN 002,6 D0 7 !TIM=INTTM1WN 0027 7 RTIM(ITIM) = TIMITTIM)
C THE CALL TO T¶AFUN,4 AL TR•A FUNR TRT BRA, 8 CHMP, Ti MEPP *MATIMEAAWI L, AC TI
C BECOMES!SN 0028 CA.Lt TRAFUN(NUMNUC,TP,TB,B,MP,TIMEPP,PTIM,RTIM,AWIGLACT, I0IMI
C BEGIN IT IM LOOP T) CALCULATE PERCENTAGE ERRORS.
TSN 0029 DO 2 ITIM = I,NTIMISN O0030 n0 4 TNUtC=1 ,MUMNUCISN 0031 4 RFLER(INUC) = (TOTUCI(ITIM,INUC) - ACT(INUCITIM))*100./
C ACT(INUC,ITIM)ISN 0032 2 WRITE (PTR,31 TIM(ITIM),(TOTUCI(ITIM,INUC),ACT(INUCtITIM),
C RELEr(INUC),INUC=1,NUMNUC)ISN 0033 3 FORMAT (1 ',9X,D13.6,IRX,E13.6,26X,E13.6,20X,Fj2.3 / '
14(40X,ElI.6,26XE13.6,20X,F12.3 / ' 'II
ISN 0034 RETURNISN 0035 END
76
LEVEL 21.8 ( JUN 74 ) OS/350 FORTRAN H
COMPILER OPTIONS - NAME= MAIN,OPT=02,LINECNT=60,SIZE=OOOOK,
ISN 0002CCC
CCCCCCCCCCCCCC
SOUPRCE,EBCOIC,NOLISTNODECK,LOAD,MAP,Nt1EDITID, NOXREFSUBROUTINE RESDNS(T,N,TR,,TB, BRANCHP,O0,QQWIGL, IDIMJ
THIS SUBROUTINE COMPUTES THE MICROCUP.IES 0 AND THEMICROCURIE-CAYS QWIGL OF N SPECIES OF A RADIONUCLIDECHAIN. EVALUATION IS AT TIME T (DAYS) IN A COMPARTMENTWITH FIRST-ORDER PEMOVAL PROCESSES WITH HALF-TIME TB(Il(rAYS) FOR THE I-TH NUCLIDE AND CONSTANT PRODUCTIONRATE P(1) (MICROCURIES/DAY). TR(I) (DAYS) IS THEQADIOACTIVE HALF-LIFE OF THE I-TH NUCLIDE. THE INITIALBURDEN IS 00(I) (MICROCURIES). BRANCH(IJ) IS THEFRACTION OF SPECIES J WHICH DISINTEGRATES TO SPECIES I,WHERE J IS LESS THAN I. THUS ALL NON-ZERO ENTRIES INBRANCH ARE BELOW THE MAIN DIAGONAL.
ALL DIMENSIONS OF SIZE 20 CORRESPOND TO N, THE NUMBER OFRADIOACTIVE SPECIES IN THE CHAIN.
IDIM IS THE MAXIMUM FIRST DIMENSION FOR THE ARRAYS ACT, P, AND AWIGL.IT CORRESPONDS TO THE MAXIMUM NUMBER OF NUCLIDES IN A CHAIN.
ISNI SN
ISNISNISNISNISNISNISN
I SNISNI SNISN
00030004
0005000700080009001000110012
0013001400150016
ISN 0017
ISN 0019
ISN 0020ISN 0021ISN 0023
ISN 0024ISN 0025ISN 0026
ISN 0027ISN 0029ISN 0030ISN 0031
REAL TR(NhTB(N),BRANCH(IDIM,IDIM),PIN),QOINIQNI ,QWIGL(N)DOUBLE PRECISION LM(20),LMRf20),D(20),C(20,20),
$ LOG2,TEMPQ,TEMPQW,EXLI,EXILIEXPFUN,EXPFIIF (T.GT.O.0) GO TO 10
DO 5 I=1,NQ(1)=00(1)OWIGL(I)=0.O
5 CONTINUEGO TO 120
10 LOG2=DLOG12.0DO)C COMPUTE DECAY AND REMOVAL CONSTANTS FROM HALF-TIMES.
DO 20 I=l,NLMR(I)=LOG2/DBLE(TR(I))LM(I)=LOG2/DBLEITB(I)) + LMR(I)
20 CONTINUEC IF TWO LM(I) ARE NEARLY EQUAL, SEPARATE THEM.C
C SKIP SEPARATION ROUTINE IF N=1.IF(N.EQ.1) GO TO 45
CC BEGINNING OF SEPARATION ROUTINE..C
N1=N-1C KODE IS A SWITCH FOR WHICH THE VALUE I MEANS ANOTHERC PASS SHOULD BE MADE.
KODE=I25 IF (KODE.NE.1) GO TO 45
KODE=OC BEGIN PASS.
00 40 K=1,NIKl=K+lDO 35 L=KI,N
C IF LM(L) AND LM(KI ARE NEARLY EQUAL, SEPARATE THEM.IF (DABS(LM(L)/LM(KI-l.ODO*.GE.1.0O-61 GO TO 35
LM(L)=LM(K)*I.O0001DOKODE=I
35 CONTINUE
77
ISN 0032 40 CONTINUEC RETURN FOR POSSIBLY ANOTHER PASS.
ISN 0033 GO TO 25C END OF SEPARATION ROUTINE.
ISN 0034 45 CONTINUECC C014PUTE COEFFICIENTS D0(), C(IJ).
ISN 0035 D(I)=P(I|/LM(l)ISN 0036 IF (N.E0.1I GO TO 60ISN 0038 DO 55 I=2,NISN 0039 0(1)=O.OISN 0040 11=1-1ISN 0041 00 50 J=1,11ISN 0042 D( I3=DI( I+BRANCH(I,J)*D(J)ISN 0043 50 CONTINUEISN 0044 O(I)=LMR(IJ*D(II/LM(I) + P(II/LM(I3ISN 0045 55 CONTINUEISN 0046 60 C(1I,=00Ot1-O(1lISN 0047 IF (N.EQ.1) GO TO 90ISN 0049 DO 85 I=2,NISN 0050 I1=1-1ISN 0051 DO 75 J=1,IlISN 0052 C(I,J)=O.OISN 0053 DO 70 K=J,I1ISN 0054 C(I,JJ=C(I,J) + BRANCHCI,K)*C(K,J)ISN 0055 70 CONTINUEISN 0056 C I, JI=C(I,JI*(LMR({I/(LM( I-LM(J)J)ISN 0057 75 CONTINUEISN 0058 C(I,I)==QO(I)-D(I)ISN 0059 DO 80 J=l,IiISN 0060 C(I,II=C(I,I)-C(I,J)ISN 0061 80 CONTINUEISN 0062 85 CONTINUE
C END OF CALCULATION OF D(I), C(I,JI.ISN 0063 90 CONTINUE
C COMPUTE 0(1), QWIGL(I), I=I,NISN 0064 DO 110 I=1,NISM 0065 TEMPQ=0.0ISN 0066 TEMPOW=0.0ISN 0067 EXLI=EXPFUN(-LM(I)*DBLE(T))ISN 0068 EX1LI=EXPFI( LM( i ),DBLETI)"ISN 0069 11=1-1ISN 0070 DO 100 J=1,I1ISN 0071 TEMPQ=TEMPQ*C(I,J)*(EXPFUN(-LM(J)*DBLE(T)I-EXLI)ISN 0072 TEMPQW=TEMPQW+C(I,J)*(EXPF1ILM(J),DBLE(T))-EXILI)ISM 0073 100 CONTINUEISN 0074 0(I)=TEMPQ+DII)*LM(IJ*EXILI*QO(I)*EXLIISN 0075 QWIGL(II=TEMPQW+D(I)*(DBLE(TI-EXILII+QO(I)*EXILIISN 0076 110 CONTINUEISM 0077 120 RETURNISN 0078 END
78
LEVEL.21.8 ( JUN 74 O 0S/360 FORTRAN: H
COMPILER OPTIONS - NAME= MAIN,OPT=O2,LINECNT=60,SZE=OOOOK,SOURCE,EBCDIC,NnLISTN1DECK,LOAD,MAP,NOEDIT,ID,.NOXR.EF
I SN 0002 DOUBLE PRECISION FUNCTION EXPFUNIT)[SN 0003 DOUBLE PRECISION TISN 0004 EXPFUN=O.ODOISN 0005 TF(T.LT.-I0.0D0O GO TO 10
TSN 0007 EXPFUN=IEXP (T)ISN 0008 10 RETURNISN 000o END
79
LEVEL 21.8 ( JUN 74 ) OS/360 FORTRAN H
COThMPILER OPTIONS - NAME= MAIN,OPT=O2,LINECNT=6OSIZE=OOOOK,SnURCE ,E BC tIC, NOLISTNODECK, LOAD, MAP, NOEDIT, ID,NOXREF
ISN 0002 DOUBLE PRECISION FUNCTION EXPFI(LM,T)ISN 0003 DOUBLE PRECISION LM,TLMT,EXPFUNISN 0004 LMT=LM*TISN 000r IF(LMT.LT.O.03DO) GO TO 10ISN 0007 GO TO 20ISN 0008 10 EXPFI=T*U((((|LMT/7.0 OO-I.OOO)*LMT/6.0D00+.ODO)
, *LMT/5.ODO-1.ODO)*,LMT/4.0DO+1.0 O I*LMT/3.ODO-1.ODOIt *LMT/2.000+*.0D0)
ISN 0009 GO TO 30ISN 0010 20 EXPFl=[I.0DO-EXPFUN(-LMT))/LMISN 0011 30 RETURNISN 0012 END
80
LEVEL 21.8 ( JUN 74 I OS/360 FORTRAN H
COMPILER OPTIONS - NAME= MAINOPT=02,LINECNT=60,SIZE=OOOOK,SOURCE,EBCDIC,NOLISTNODECKLOAD, MAP,NOEDIT, IDNDXREF
ISN 0002 SUBROUTINE TRAFUNIN, TRTB, BRANCH,MP,TIMEPP, MA,TIMEAAWIGLACT,* IDIM)
C TRAFUN SUGGESTS TRANSFER FUNCTION.C WE ARE CONSIDERING N RADIOACTIVE SPECIES IN A CHAIN IN A BIOLOGICALC COMPARTMENT. TRII) AND TB(Il ARE THE RADIOACTIVE HALF-LIFE (DAYS)C AND BIOLOGICAL HALF-TIME (DAYS), RESPECTIVELY, OF THE I-TH SPECIESC IN THE COMPARTMENT. BPANCH(I,J) IS THE BRANCHING RATIO OF SPECIESC J TO SPECIES I, WHERE J IS LESS THAN I. THE INFLOW RATE OF EACHC SPECIES IS GIVEN AS A DISCRETE FUNCTION OF TIME BY THE ARRAYS
C TIMEP (DAYS) AND P (MICROCURIES/DAY). P(I,KP) IS THE INFLOW RATEC OF SOECIES I FROM TIMEP(KP) TO TIMEP(KP+I|, AND P(I,MPl IS THEC RATE AT TIMES SUBSEQUENT TO TIMEP(MP). AWIGL(IKAI IS THEC CUMULATED ACTIVITY (MICROCURIE-DAYSI IN THE COMPARTMENT UP TOC T!MEA(KA). THE TIME ARRAYS MUST BE ARRANGED IN INCREASING ORDER.C ACT(I,KA) IS THE ACTIVITY FOR NUCLIDE I (MICROCURIES) IN THEC COMPARTMENT AT TIMEA{KA).CCCCC
ISN 0003ISN 0004ISN 0005
CCCCCC
ISN 0006
ISN 0007ISN 0008ISN 0009ISN 0010ISN 0011ISN 0012ISN 0013ISN 0014I SN 0015
CC
ISN 0016CCC
IDIM IS THE MAXIMUM FIRST DIMENSION FOR THE ARRAYS ACT, P, AND AWIGL.IT CORRESPONDS TO THE MAXIMUM NUMBER OF NUCLIDES IN A CHAIN.
THE MAXIMUM DIMENSION 20 FOR THE ARRAYS PTEMPATEMP AND AWTEMP CORRES-PONDS TO N (THE NUMBER OF RADIOACTIVE NUCLIDES IN THE CHAIN).
REAL TR(N),TB(N),BRANCH(IDIM,IDIM|,ACT(IDIM,365)REAL TIMEP(30),P(IDIM,30),TIMEA(MA),AWIGL(IDIM,3651REAL PTEMP(20),ATEMP(20),AWTEMP(20)
FOR EACH TIMEP(KPI AND THE CORRESPONDING COLUMN P(*,KP) OFRATES, USE RESID ITERATIVELY TO CALCULATE THE CONTRIBUTIONTO AWIGL(*,KA) AT TIME TIMEA(KA). FIRST INITIALIZE AWIGL AND ACTTO ZERO.
COMMON /INDEP/ A,ASURG,DSUBG,MSUBBTAU8EFTAUMLKTAUESTAUGR,* TAUPH,TAURD,TAURG,TAUSP,UVSUBCSMALLD,RHO
COMMON /SOURCE/ FO(15)CALL ZEROM(IDIM,MAAWIGL)CALL ZEROM(I!DIM,MA,ACTI00 25 KP=1,MP
DO 10 I=l,NPTEMP(I)=P(IKPI
10 CONTINUEDO 20 KA=I,MA
CALL ZEROV(N,ATEMP)IF KP=l, SET INITIAL TOTAL ACTIVITY IN THE SYSTEM EQUAL TO THATDETERMINED BY FOIl1, 1=1 TO N AND S$,S2,$3,$4.
IF (KP .NE. 1) GO TO 40
CALCULATE TOTAL INITIAL ACTIVITY (MICROCURIES) IN RAGTIME COMPARTMENTSE,SG AND R.
TZERO = 0.0CALL SVAL(TZERO,SIS2,S3,S4)00 41 I=1,N
41 ATEMP(I) = (fSIS+2)*A + (S3+S4)*ASUBGI * FOIl)40 CALL ZEROV(N,AWTEMP)
Tl=TIMEPIKPIIF (Tl.GT.TIMEA(KAII GO TO 20IF IKP.LT.MP) T2=TIMEP(KP+1)IF (KP.EQ.MP) T2=TIMEA(KA)
I SNISNI SNI SNI SNISNISNISNISN
001800190020002100220023002400260028
81
ISN 0030 CALL RESID(NTRTBBRANCH, ATEMP,PTEMP,T1,T2,$ AWTEMP,TIMEA{KA),[DIM)
ISN 0031 DO 15 I=1,NISN 0032 AWIGL(I,KA)=AWIGL(!,KAI+AWTEMP(1)!SN 0033 ACT( IK) = ATEMP(II + ACT(I,KA)ISN 0034 15 CONTINUEISN 0035 20. CONTINUEISN 0036 25 CONTINUEISN 0037 RETURNISN 0038 END
82
LEVEL 21.8 ( JUN 74 1 OS/360 FORTRAN H
CCMPILER OPTIONS - NAME= MAIN,ODT=02,LINECNT=60,SITZE=OOOK,SUPCE, E BCO C, NOL I ST ,NOECK,L OAD, MAP, NOEDI T, ID,NOXREF
ISN 0002 SUBROUTINE ZEPOMIN,M,.4ISN 0003 ODIPENSIGN A(N,M,!SN 0004 00 10 I=INISN 0005 [) 10 J=1,MISN 0006 A(1 7 J)=O.O!SN 0007 1O CONTINUEISN 0008 RETURN1SN 0009 END
83
LEVEL 21. 8 ( -JUN 74 OS/ /S16 FORTRAN H
CO-MPILER OPT•rqNS - NAME= MATN,(OPT=02,LINECNT=60,SIZE=OOOOK,S0IJcCE,EBCD!C,NOLIST,NODECKLOAD,MAP,NOEDIT,ID,NOXREF
!SN 30002 SUBROUTINE ZErOV(N,Vi!SN 0003 01MENSION V(N)TSN 0004 DO 10 I=1,N!SN 0005 10 VCI=0.OI5J' 0006 RETUPNI SN 0007 END
84
LEVEL 21.8 i JUN 74 1 OS/360 FORTRAN H
COMPILER OPTIONS - NAME= MAIN,OPT=02,LINECNT=60,SIZE=OOOOK,SOURCEFBCDIC,NOLIST,NODECKLOAD,MAP,NDEDITIDNOXREF
ISN 0002 SUBROUTINE RESIOIN,TR,TBBRANCH,A,P,T!,T2,AW,T,IDIM)C COMPUTES MICROCURIE-DAYS RESIDENCE AW(Il OF THE I-TH RADIOACTIVEC SPECIES IN A CHAIN OF N NUCLIDES. PARAMETERS TR,TB,ANDC BRANCH ARE AS IN SUBROUTINE TRAFUN. INPUT IS A PULSE VECTORC P (MICROCURIES/DAY) FROM TIME TI TO T2 (DAYS). INITIAL VECTORC OF ACTIVITIES IS A (MICPOCURIES), AND A IS UPDATED TO SHOWC FINAL ACTIVITIES. AW IS EVALUATED AT TIME T. IN CASE T ISC LESS THAN TI, S IS UNCHANGED AND AW IS ZERO.C IDIM IS THE MAXIMUM FIRST DIMENSION OF THE ARRAY BRANCH.C THE MAXIMUM DIMENSION 20 FOR THE ARRAYS AI,PI AND AWL CORRESPONDS TOC N (THE NUMBER OF NUCLIDES IN THE CHAIN).
ISN 0003 REAL TR(NbTB(N),A(N),P(N),AW(N),BRANCH(IDIM,IDIM)ISN 0004 REAL At(201,PIf20l,AWL(20lTSN 0005 CALL ZEPOV-N,PIIISN 0006 IF (TI.GT.T2) GO TO 20!SN 0008 TTEMP=AMAXI(O.O,AMIN1(T,T2 )-TI)ISN 0009 CALL RESONS(TTEMP,N,TRtTB,BRANCH,PA,A1,AWIDIM)ISN 0010 DO 10 I=I,NISN 0011 10 A(I)=A!(1lISN 0012 IF (T.LE.T2) GO TO 30ISN 0014 TTEMP=T-T2ISN 0015 CALL RESDNS(TTEMP,N,TR,TB, BRANCHP1,AA1,AWIIDIM)ISN 0016 DO 15 I=I,NISN 0017 A(II=Al(I)ISN 0018 AW(Il=AWtI)+÷AWl(IISN 0019 15 CONTINUETSN 0020 20 CONTINUEISN 0021 TTEMP=AMAXI(0.O,T-T1)ISN 0022 CALL RESDNS(TTEMP,N,TR,TB,BRANCHPI,A,A1,AW)ISN 0023 DO 25 I=ItNISN 0024 25 A(II=Alf11TSN 0025 30 RETURNISN 0026 END
85
LEVEL 21.8 ( JUN 74 ) OS/360 FORTRAN H
COMPILER OPTIONS - NAME= MAINOPT=02,LINECNT=bO,SIZE=OOOOK,SOURCE,EBCOICt NOLIST, NODECK, LOAD,MAP,NOED IT, 10, NOXREF
TSN 0002 SUBROUTINE SVAL(T,SI,S2,S3,S4).CC RETURNS VALUES OF INTERCEPTION FRACTIONS SI,S2,$3,S4 AT TIME T (DAYS).C
ISN 0003 COMMON /HRVST / HARTIM(30),NHARVISN 0004 COMMON /EMERG/ EMERGE(30)ISN 0005 OOUBLE PRECISION HARTIMISN 0006 REAL M,MFOSI!SN 0007 MFOS.1 = 1.0ISN 0008 SLO=0.00075ISN 0009 ATAU=1.24E-4ISN 0010 S1=0.0ISN 0011 DO I I= 1,NHARVISN 0012 !F(T.GE.EMERGE(I).ANOD.T.LT.HARTIMII)I GO TO 2ISN 0014 1 CONTINUE
C IF T DOES NOT LIE BETWEEN EMERGE(I) AND HARTIMLI) FOR ANY 1=1 TOC NHARV, THEN SI=0.
ISN 0015 GD TO 3ISN 0016 2 TO=EMERGE(I)ISN 0017 M = MFOSI*(I.O-EXP(-ATAU*(T**2-TO**2)I)ISN 0018 S1= SLO*(M**0.545)*250.0ISN 0019 3 CONTINUEISN 0020 IF(S1.LE.1.0 GO TO 4
C SI SHOULD NCT EXCEED 1.ISN 0022 WRITE(3,5) SIISN 0023 5 FORMAT('OS1 VALUE (',ElO.3,') TOO LARGE')ISN 0024 STOPISN 0025 4 S2= 1.0 -SIISN 0026 S3= 0.25ISN 0027 S4= 0.75ISN 0028 RETURNISN 0029 END
86
LEVEL 21.8 ( JUN 74 ) OS/36) FORTRAN H
COMPILER OPTIONS - NAME= MAIN,OPT=O2,LINECNT=60,SIZE=OOOOK,SOURCE,EBCDIC,NOLISTNODECK,LOAD,MAP,NOEDIT,ID,NOXR EF
ISN 0002 SUBROUTINE HARVSTCC DEFINES HARVEST TIMES (DAYS)C
C HARTIM(T), 7=1 TO NHARVCC AND COMMUNICATES THESE VALUES TO SURROUTINE CALC VIAC COMMON BLOCK /HRVSTI.C
I SNISNI SNI SNISNI SNI SNI SN
00030004000500060007000800090010
DOUBLE PRECISION HARTIMCOMMON /EMERG/ EMERGE(30)COMMON /HRVST/ HARTIM(301,NHAP.VNHARV = 1EMERGE{IJ = 70.HARTIM(1) = 175.000RETURNEND
87
LEVEL 21.8 1 JUN 74 OS/36) FORTRAN H
COMPILER OPTIONS - NAME= MAIN,OPT=O2,LINECNT=60,SIZE=OOOOK,SOURCE ,E BCDI C, NOLIST ,NODE CK, LOADMAPNOEDIT, 10, NOXREF
ISN 0002 SUBROUTINE TIMDEP (TTAUPTP
C
C * SUBROUTINE T I M D E P *
CC TIMDEP RETURNS TAUT)TfI) = TRANSFER COEFFICIENT FROM P TP T (SUBSUR-C FACE SOIL POOL TO PLANT INTEPIORI FOR NUCLIDES NAMNUCII(I,1=,NUMNLC.C TAUPT IS BASED ON THE TIME INTERVAL CONTAINING T.
C.ISN 0003 DIMENSICN TAUPT(.5)
CISN 0004 COMMON /HRVST/ HARTIM(301,NHARVISN 0005 COMMON /EMERG/ EMERGE(30O1 SN 0006 COMMON /CROPS/ MFODISN 0007 COMMON /NAMES/ NAMNUC(15)ISN 0008 COMMON /NUMeRS/NUMNUC
ISN 0009 COMMON /DEP/ LAMRR(15),FSUBM(151,TAUEXC(15),FSUBF(15),BSUBIV(15)tSN 0010 COMMON /INDEP/ AASUBGDSUBG,MSUBBTAU8EFtTAUMLK,TAUESTAUGR,
STAUPHTAURDTAURG,TAUSP,UVSUBC,SMALLD,RHO
ISN 0011 DOUBLE PRECISION NAMNUC,HARTIMISM 0012 REAL LAMRR,MSURB,MFO
CISN 0013 ATAU = 1.24E-4
CC FIND THE TIME INTERVAL (EMERGEWJ) TO HARTIMIJ) ) CONTAINING T.
ISN 0014 00 10 J=1,NHkRVISM 0015 IF IT *GE. EMERGEIJI *AND. T *LT. HARTIMIJ}) GO TO 20ISN 0017 10 CONTINUE
CC T IS NOT IN ANY OF THE GIVEN TIME INTERVALS (EMERGEIJ) TO HARTIMIJ) I
ISM 0018 01 15 I=1,NUMNUCISN 0019 TAUPT(I) = 0.0ISN 0020 15 CONTINUE
CC FND OF PROCESS IF T IS OUTSIDE OF THE GIVEN TIME INTERVALS.
ISM 0021 GO TO 30CC T IS IN THE TIME INTERVAL EMERGE(JI TO HARTIM(J). TO IS EQUALC TO THE EMERGENCE TIME FOR THE CROP INTERVAL.
ISN 0022 20 TO = EMERGEIJ)CC CALCULATE TAUPT FOR EACH NUCLIDE IN THE CHAIN.
ISN 0023 DO 25 I=1,NUMNUCISN 0024 UDOT = 2.0 * ATAU * T * EXP(-ATAU * (T**2 - TO**2))ISM 0025 TAUPT(I) = (MFO * UDOT * BSUBIV(I)) / (10000.0 * A * SMALLD*RHO)ISN 0026 25 CONTINUE
CISN 0027 30 RETURNISN 0028 END
88
LEVEL 21.8 ( JUN 74 O OS/363 FORTRAN H
COMPILER OPTIONS - NAME= MAIN,OPT=O2,LINECNT=60,SIZE=OOOOK,SOURCE,EBCDIC,NDLIST,NODECK,LOAD,MAP,NOEDIT,ID,NOXREF
ISN 0002 SUBROUTINE OUERY(T, TANS)CC IS CALLED BY SUBROUTINE CALC TO DETERMINE WHETHER OR NOT TIME T (DAYS) ISC A HARVEST TIME. THE VALUE OF IANS IS SET TOCC I IF T IS A HARVEST TIME,C 0 IF NOT.CC OUERY SEARCHES THE ARRAY HARTIM, WHICH IS DEFINED BY SUBROUTINE HARVST,C IN ORDER TO DETERMINE WHETHER OR NOT T IS A HARVEST TIME, I.E. WHETHERC OR NOT T= HARTIM(J) FOR SOME J=I TO NHARV.
ISN 0003 COMMON /HRVST/ HARTIM(30),NHARVISN 0004 DOUBLE PRECISION HARTIM,TISm 0005 IANS = 0ISN 0006 DO 1 J=I,NHARVISN 0007 IF (T.EQ.HARTIM(Jfl IANS=.lISN 0009 IF(IANS.EQ.1) GO TO 2ISN 0011 1 CONTINUEISN 0012 2 RETURNISN 0013 END
89
LEVEL 21.8 ( JUN 74 ) OS/)60 FORTRAN H
COMPILER OPTIONS - NAME= MAINOPT=02,LINECNT=60,SIZE=OOOOK,SOURCE,EBCOIC,NOLIST,NIDECK,LOADMAPNOEDIT, IDNOXREF
CC RAGTIMEC ...........-- - ---
CC PROGRAM AUTHORS : J.C.PLEASANT, L.M.MCDOWELL-BOYER, AND G.G.KILLOUGH
C HEALTH AND SAFETY RESEARCH DIVISIONC OAK RIDGE NATIONAL LABORATORYC OAK RIDGE, TENNESSEE 37830CC ------------------------------------------------- -- --
CC PTR IS USEC TO REPRESENT THE UNIT NUMBER ASSOCIATED WITH THE LINEC PRINTER, RCP IS USED TO REPRESENT THE UNIT NUMBER ASSOCIATED WITHC THE CARD READER.C
ISN 0002 COMMON /!ODFV / PTR.RDRISN 0003 INTEGER PTR, RDRISN 0004 ROR = 5ISN 0005 PTR = 6
CiSN 0006 IFLAG = 0
CC [FLAG IS A PARAMETER PASSED IN SUBROUTINE INPUT WHICH DIRECTS
C THE FLOW OF THE PROGRAM. IF IFLAG IS SET TO ZERO, THE ENTIREC SUBROUTINE IS EXECUTED AND IFLAG IS SET EQUAL TO ONE. THISC ENABLES A BRANCH TO BE MADE AROUND THE PORTION OF CODE THAT W4PUTSC THE NUCLIDE INDEPENDENT PARAMETERS ON SUCCESSIVE CALLS FOR THEC VARIOUS CHAINS BEING STUDIED.C
ISN 0007 READ (RDR,1) NCHAINISN 0008 10 FORMAT (1OX,12)
CISN 0009 00 20 1 = I,NCHAINISN 0010 CALL INPUT iIFLAGIISN 0011 CALL OUTDATISN 0012 CALL CALCINISN 0013 CALL HARVSTISN 0014 CALL CALCISN 0015 CALL CHECKISN 0016 20 CONTINUE
CISN 0017 STOPISN 0018 END
91
Job Control Language (JCL) for RAGTIME
Job control language varies from one computer installation to
another. For execution of RAGTIME on the IBM 360/91 at Oak Ridge
National Laboratory, the following JCL arrangement has been used:
//jobname JOB (charge no.),'X-1O 7509 PLEASANT'//*CLASS CPU91=44S,I0=2.8,REGI0N=270K/*ROUTE XEQ CPU91
II EXEC FORTlnCLG,REGION.O2-7IOUK,mAKM.GU='EU=-I//!ORT.SYSIN DD _*
source decks (RAGTIME MAIN and subroutines)
//LKED.GEAR DD DSN=T.GGK05716.GEAR,DISP=SHRUNIT=SPDA,// DCB=(RECFM=FB,LRECL=80,BLKSIZE=800)//LKED.SYSIN DD m
INCLUDEGEAR
//GO.FTO3FOO1 DD SYSOUT=A,DCB=(RECFM=VBA,LRECL=137,BLKSIZE=1000)//GO. FTO1FOO1 DD *
data deck
/1
The underline (_) is used to indicate a space. The JCL shown above
makes use of compiled code for GEAR stored in the system as a cata-
logued data set and made available to the Linkage Editor through the
JCL statements comprehended by the brace. If the subroutines of the
GEAR package are to be compiled along with RAGTIME, they should be
included with the source decks and the JCL statements in the brace
deleted. The additional compilation time would require that the limits
on the CLASS card be revised. We note also that other sets of input
data (e.g., radionuclide chains with more than two species or multiple
problems within one job) will require longer running times. Moreover,
the running times will vary greatly with the model of IBM system and
other local factors.
APPENDIX B
SAMPLE RUN OF THE RAGTIME CODE
95
APPENDIX B
Output from a sample run of the currently implemented version of
RAGTIME is provided in this appendix. Values for all state variables
are listed, and concentrations in grains, milk, and beef are plotted
versus time in Fig. B.1, B.2, and B.3. A number of parametric values
had to be specified and options chosen to complete this run (see
Table B.1). Table B.2 exhibits a listing of the data cards used for
the sample run.
Following is a brief description of values and options specified
for the sample run conducted, which considers a chronic deposition term
(Fi) of 1 pCi 9 °Sr per m2 per day, beginning 70 days prior to emergence
of the grain. Compartment Ei, representing direct contamination of the
surface of above-ground food crops through interception of depositing
radionuclides, was considered to consist only of grain crops. In doing
so, the time-dependent intercepting efficiency of the edible portion of
the crop, the grain, was modeled using empirical values describing
projected surface area as a function of plant mass, rather than using
the geometric approach (see Sect. 3 discussion). Both empirical values
and plant growth curves were obtained mainly from work documented by
Miller' of the Stanford Research Institute. An equation describing the
mass m of the grain per plant (grams/plant) was adopted from Miller; at
any time t > to,
m = m [ t 2 t0 , (B.1)
0where mf = final mass of grain at harvest (grams/plant); t0 = time of
emergence of grain (days), and aT = growth coefficient (day- 2 ). All of
these input parameters represent averages for a number of grain varie-
ties. For the sample run of RAGTIME, the following values were used:
0mf = I gram/plant
tO = 70 days
a = 1.24 x 10-4 day-2
20
<16(9
z•12
ui0z0UuJ 40
0
ORNL-DWG 79-19515
'.0
0 40 80 120 160 200TIME FOLLOWING'INITIATION OF DEPOSITION (days)
Fig. B.1. Predicted concentrations of 9 °Sr and 90Y in grain vs time (1 pCi m-2 day- 1
deposition of 9 0 Sr).
• • • II • m A mA qA IA P J •• .2(OlNL-UW5 79-19514. 1.2 I
i.0'
_0.8 9 0 Sr
0
4O.6
z0
--
I--L' 0.42"0)9
_ _ _
z 00 40 80 120 160 200
TIME FOLLOWING INITIATION OF DEPOSITION (days)
Fig. B.2. Predicted concentrations of 9"Sr and 9 0Y in milk vs time (1 pCi m-2 day-'deposition of 9 0Sr).
0.048
1_ 0.040i~i
m0.032
z0
# 0.024
zo 0.016z0u
0.008
0 0
ORNL- DWG 79-19513
0 40 80 120 160 200TIME FOLLOWING INITIATION OF DEPOSITION (days)
Fig. B.3. Predicted concentrations of 9°Sr and 9 0Y in beef vs time (1 pCi m-2 day- 1
deposition of 9 °Sr).
Table B. 1. Values of parameters and other quantities used in samplerun of RAGTIME for the 90 Sr, 9 0Y decay chain'
Reference or sectionFORTRAN Specific of report containing
Parameter name for Value used discussion
A A 1,000 m2
10,000 m2
4
4Ag
aT
B2 1
B.Iv
ASUBG
ATAU
B(2,1)
grains 1.24 x 10-4 day- 2 App. B
1 (= radioactive branchingradio from 9 0 Sr to 9 0Y)
Sect. 6.1
In
BSUBIV(I) 0.290 for 9 0 Sr,0.00430 for 90Y
2, 3
d SMALLD
DSUBG
20 cm 4
40g
Fi(t)
(Ff)i
(F m)i
0.15 kg m-2
F(I ,T)
FSUBF(I)
FSUBM(I)
F1 (t) =1 pCi m-2 day-'F2 (t) = 0 pCi M-2 day-'
3.0 x 10-4 for 9 0 Sr
5.8 x 10-3 for 90Y
2.4 x 10-3 for 9 °Sr2.0 x 10-5 for 9 0Y
App. B
2
2
Table B.1 (continued)
Reference or sectionFORTRAN Specific of report containing
Parameter name for Value used discussion
LAMRR(I) X, = 6.66 x I0-s day- 61X2 = 0.26 day-'
o • -•(t2- to)1m grains mf L - e J , t > to, 1
where t represents time (days)
Mb MSUBB 200 kg 40
Mf MFOS1 grains 1 g/plant App. B0
Mf MFO grains 250,000 g App. B
nL NSUBL grains 0.455 1
p RHO soil 1.4 g/cm- 3 4
S1 S1 grains 0.00075 m°-5 45w 1
S 2 $2 grains 1 -5 Sect. 3.1
S3 S3 pasture 0.25 Sect. 4.1
S4 S4 pasture 0.75 Sect.. 4.1
SL SL grains 0.00075m- 0 . 45 5 1
0SL SLO grains 0.00075 1
Table B.1 (continued)
Reference or sectionFORTRAN Specific of report containing
Parameter name for Value used discussion
Tbeef
Te,s
Texc
T *
(Tg,b)i
(g,c i
Tg,r
T milk
Tp,h
(rp~t)i
T r,d
Tr,g
T I Ps,p
thtb
TAUBEF
TAUES
TAUEXC(I)
TAUGB(I)
TAUGC(!)
TAUGR
TAUMLK
TAUPH
TAUPT(I)
TAURD
TAURG
TAUSP
TIMBH
3.81 x 103 day- 1
0.0495 day'1
2.0 x 10-3 day-' for 9 °Sr,2.0 x 10-3 day-' for 90Y
Vc/(AgDg) 6.67 x 10-3 day-'
[(F f)i(exc)iVc]/Dg9
m(Fm)i TmilkVc /Dg
0.0495 day- 1
2 day-'
1.096 x 10-4 day-'0.
[MfU(t)Biv]/(ZO,O00 x Adp)
1.096 x 10-4 day-'
2.74 x 10-i day-'
6.93 x 10-4 day-'
20 days
4
4, 7
4
Sect. 4.1
Sect. 4.2
Sect. 4.2
4, 7
4
4, 8
App. B
4, 8
Sect. 4.1
Sect. 2.1
5
I-
grains
Table B.1 (continued)
Reference or sectionFORTRAN Specific of report containing
Parameter name for Value used discussion
th TIMCH 2 days App. Bc
to TO grains 70 days App. B
U U 5.5 liters 4
2-a (t 2 - to)
0(t) UDOT grains 2a te , t > to, 1
where t represents time (days)
Vc VSUBC 10 kg day-' 4
w W grains 250 plants m-2 App. B
aThe subscript i used with parameter names refers to the ith nuclide (i 1 for 9 °Sr,i = 2 for 90 y).
ro
103
Table B.2. Format of input parameters for RAGTIME
Sp9o 0.0
o.0.'0
O.Ok) .IOEO0, 01 0 . OEO0
SN(0 Y90 1.011. 365I. AMPR Skb906.66 E-05FS-_BN S.P .90?4 E-03FS(JHF SR903.0 E[-0 4BSUr RT vSPq02. 9 E-601TAUF. XCSP90().002LANIPP Y90 2.60 E-01FSORIM Y90 2.0 E-05FSIUOF Y90 5.8 E-03RSU$IVY9O 4.3 E-03TAOEXCY90 0.002A 1 1000.0ASL1G 10000.0PS.IU3F 0 1DSUIZ, 0.15S', A 1) 21.0
Dl 0.?5P2 0.2503 1.0
MSHIB P200.RHO I .14T AUIBF F 0.00381T A .1MIt. K P .0T AUtRM 0.3T A U ClM 1.0TA lS .0.095TAU(;P 0.0495TAUPD 0.0001096T AIpD 0.0001096TAUPL71 0.000027I 4TAuSP 0.000693T AU T .00U 5.5V 0.25VSI.ic 10.0VS1IRS 280000000.0LAMEH I.OE+I01LIANIH I.OE+01LAMHH 5.0E-0?L A M, C H 5.0 - 0 1
104
The specified values were derived from a minimum of data, for the grain
of one variety of wheat, and thus may not be the best values to use for
other simulations.
We have made use of Eq. (B.1) in deriving time-dependent values
for the interception fraction Si and a normalized version of this
equation in the derivation of values for the transfer coefficient
(Tprt)i. The fraction S, may-be viewed as the ratio of the projected
surface area (m2 ) of the grain to the area (M2 ) of the land on which
the crop is grown. Thus, Sl is related to-the specific area,
S projected surface area of grain (m2 )L mass of grain (grams)
by the equation
S,= S Lmw (B.2)
where m = mass of grain per plant (grams), and w number of plants per
square meter of land. The specific area, SL' may be fitted to an
equation of the form
-n LSL = •m(B.3)
0in which S[ and nL are empirical constants and m is the time-dependent
mass of the grain per plant as given in Eq. (B.1). This model is
adopted from ref. 1 (p. 177). Using the values S = 0.00075 and nL
0.455 from this document, the value of S, is calculated from Eqs. (B.2)
and (B.3) to be
S, = 0.00075 mI. 5 45 w
For the root uptake compartment Ti, again only grains were con-
sidered in the sample run. The rate (microcuries day-) at which
radioactivity is absorbed by plant roots is represented in Eq. (2.9) byI
105
the term (Tpdt)iPi, where Pi represents the radioactivity (pCi) present
in the subsurface soil pool associated with one man's food supply. As
in our discussion of Eq. (B.1), it follows that the total mass (grams),
as time t > tO, of grain in compartment T. is given by M0U(t), where
2 2
U(t) 1 - ea - 0) (B.4)
0and Mf = Lotal mass (grams) of crop at harvest time grown on land
associated with one man's food supply. The rate of root absorption of
radioactivity, (Tp,t )Pi.(microcuries day-') is assumed to be the prod-
uct of the rate of increase of grain mass MfU(t)(grams day-') and the
radioactivity concentration in grain (microcuries gram-'), the latter
quantity being approximated by
B. P. B. P.Iv i iv i (B.5)
mass (grams) of soil 10,000 x Adpin compartment Pi
where B.iv concentration of nuclide i per unit fresh weight in plant
(pCi kg-') divided by concentration of nuclide per unit dry weight in
soil (pCi kg-1); A = soil area used for crop production, chosen as 103
m2 here; d = plow depth, assumed to be 20 cm; and p = soil density,
assumed to be 1.4 g cm-3 (dry weight). This derivation then leads to
the equation
M' U(t) B.f iv(B6(tp t i= 10,000 x Adp (B.6)
where
2 2-aT (t - t 0 )
ý L ) = 2a L-e kin. /)
106
Factors describing growth rate [i.e., U(t) and a ] were again derivedfrom empirical data available for grains' and are consistent with those
used in estimating interception of airborne radionuclides by grains.
Values of Biv (see Table B.1) were derived from empirically obtained
data for the elements Sr and y. 2 , 3
For compartments B. and C., respectively representing beef and
milk concentrations of 9 °Sr and 90Y, all parameters were defined and
assigned values in Sect. 4 of this document, with the exception of
(Texc~i5 (F f)i , (Fmi, tb, and th A constant value of 0.002 day-,
adopted from TERMOD, 4 has temporarily been assigned to ( exc)i. This
value for ( exc)i represents the fractional weight gain per day for amature steer, and thus implies that dilution of the elemental concen-
tration is due only to growth. In this sense, it represents the lower
limit for ( exc)i, and thus may underestimate loss from the beef com-
partment because metabolic turnover, which may be element-specific, is
neglected. Values for Ff and Fm (Table B.1) were taken from a review
of literature concerning uptake of these elements by cattle and subse-
quent transfer to meat and milk, respectively.
The holdup times for compartments Bi and Ci were specified using
values either given in, or derived from, the U. S. Nuclear Regulatory
Commission's (USNRC) Regulatory Guide 1.109 (October, 1977).1 The
assumed time between slaughter and consumption of beef (tb) was 20h b
days, and between milking and milk consumption (t ) was 2 days, thelatter representing one-half of the total time given for transfer from
feed, through milk, to man. For compartments Ei and T., holdup times
were not specified, but rather the concentrations after harvest are
printed at each output time so that the value of this parameter is left
to the user's discretion.
107
REFERENCES - APPENDIX B
1. C. F. Miller, Operation Ceniza-Arena: Retention of Fallout Parti-cles from Volcan Irazu (Costa Rica) by Plants and People, PartThree, SRI-MU-4890, Stanford Research Institute, Menlo Park,California (December 1967).
2. R. E. Moore, C. F. Baes III, L. M. McDowell-Boyer, A. P. Watson,F. 0. Hoffman, J. C. Pleasant, and C. W. Miller, AIRDOS-EPA: AComputerized Methodology fnr Estimating Environmental Concentra-tions and Dose to Man from Airborne Releases of Radionuclides,ORNL-5532, Oak Ridge National Laboratory, Oak Ridge, Tennessee(June 1979).
3. E. M. Romney, J. W. Neal, H. Nishita, J. H. Olafson, and K. H.Larson, "Plant Uptake of Sr90, Y91, Rul06, CS137, and Ce144 fromSoils," Soil Sci. 83, 369-376 (1957).
4. R. S. Booth and S. V. Kaye, A Preliminary Systems Analysis Modelof Radioactivity Transfer to Man from Deposition in a TerrestrialEnvironment, ORNL/TM-3135, Oak Ridge National Laboratory, OakRidge, Tennessee (October 1971).
5. U. S. Nuclear Regulatory Commission, Calculation of Annual Dosesto Man from Routine Releases of Reactor Effluents for the Purposeof Evaluating Compliance with 10 CFR Part 50, Appendix I, Regula-tory Guide 1.109, Revision 1, USNRC, Washington, D. C. (October1977).
6. D. C. Kocher, Nuclear Decay Data for Radionuclides Occurring inRoutine Releases from Nuclear Fuel Cycle Facilities, ORNL/NUREG/TM-102, Oak Ridge National Laboratory, Oak Ridge, Tennessee(August 1977).
7. J. D. Zimbrick and P. G. Voilleque, ed., 1967 CERT Progress Re-port, Controlled Environmental Radioiodine Tests at the NationalReactor Testing Station, Progress Report Number Four, IDO-12065,Idaho Falls, Idaho (December 1968).
8. R. G. Menzel, "Factors Influencing the Biological Availabilityof Radionuclides for Plants," Fed. Am. Soc. Exptl. Biol. Proc. 22,1398-1401 (1963).
109
NUREG/CR-1196ORNL/NUREG/TM-371
Distribution Category RH
INTERNAL DISTRIBUTION
1.2.3.4.5.6.7.8.9.10.11.12.13.14.15.
16-19.20.21.
22-25.
C.B.J.
R.S.D.K.E.D.R.C.F.G.F.S.G.D.C.L.
F. Baes IIIA. BervenCarney0. ChesterJ. CotterE. DunningF. EckermanL. EtnierE. FieldsH. GardnerT. Garten, Jr.F. HaywoodS. Hill0. HoffmanV. KayeG. KilloughC. KocherA. LittleM. McDowell-Boyer
26.27.28.29.30.31.3233.34.35.36.37.38.
39-40.41.
42-43.44.45.46.
H. R. MeyerC. W. MillerT. MyrickF. R. O'DonnellR. V. O'NeillD. C. ParzyckP. S. RohwerG. SchwarzR. W. ShorJ. E. TillC. C. TravisA. P. WatsonJ. P. WitherspoonCentral Research LibraryORNL Y-12 Technical LibraryLaboratory Records DepartmentLaboratory Records, ORNL-RCRSIC LibraryORNL Patent Office
EXTERNAL DISTRIBUTION
47. D. Kaul, Science Applications, Inc., 1 Woodfield Place Building,1701 East Woodfield Road, Suite 819, Schaumburg, IL 60195.
48. Y. C. Ng, Lawrence Livermore Laboratory, University of Califor-nia, Livermore, CA 94550.
49. J. C. Pleasant, Department of Mathematics, East Tennessee StateUniversity, Johnson City, TN 37601.
50. F. W. Whicker, Department of Radiology and Radiation Biology,Colorado State University, Fort Collins, CO 80523.
51-52. Technical Information Center, ORO, Oak Ridge, TN 37830.53. Office of Assistant Manager, Energy Research and Development,
DOE-ORO, Oak Ridge, TN 37830.54-233. Given distribution according to NRC Category RH.
*U.S. GOVERNMENT PRINTING OFFICE: 1980-640-245/085
